JEE (Main + Adv)

Exercise Sheet

Class XI

MATHS
Parabola
PARABOLA
 INDEX
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 SOLVED EXAMPLES
Ex.1 Find the equation of the parabola whose Ex.2 Find the equation of the parabola whose
focus is (1, 1) and the tangent at the vertex is axis is parallel to the y-axis and which
x + y = 1. Also find its latus rectum. passes through the points (0, 4), (1, 9),
Sol. The directrix is parallel to the tangent at (–2, 6). Also find its latus rectum.
the vertex V. Sol. As the axis is parallel to the y-axis, it will
∴ The directrix will be of the form x + y = λ be x – α = 0 for some α and the tangent to
...(i) the vertex (which is perpendicular to the
Now, V is the foot of the perpendicular x-axis) will be y – β = 0 for some β.
from S (1, 1) to the line x + y = 1.
Hence the equation of the parabola will be

T
Let V = (α, β), then α + β = 1 ...(ii)
of the form (x – α)2 = 4a (y – β) ...(i)
β −1
and (–1) = – 1, i.e. α = β ...(iii)
x–α=0

IN
α −1
1
Solving (ii), (iii) we get α = β = . Y=0
2
⎛1 1 ⎞ x=0
So V = ⎜ , ⎟
⎝2 2⎠ y–β= 0

PO
directrix
When α, β, a are unknown constants, 4a
V S(1, 1) being latus rectum.
M
(i) passes through (0, 4), (1, 9) and (–2, 6).
So (0 – α)2 = 4a (4 – β),
directrix i.e., α2 = 4a(4 – β) ...(ii)
x+y=1 and (1 – α)2 = 4a (9 – β)
Let M = (x1, y1). As MV = VS, V is the
R
i.e., 1 – 2α + α2 = 4a (9 – β) ...(iii)
middle point of MS.
and (–2 – α)2 = 4a (6 – β)
x + 1 1 y1 + 1 1
∴ 1 = , = i.e., 4 + 4α + α2 = 4a (6 – β) ...(iv)
2 2 2 2
EE

(iii) – (ii) ⇒ 1 – 2α = 20a ....(v)

x1 = 0, y1 = 0. So, M = (0, 0)
As M is on the directrix, (0, 0) satisfies (i) (iv) – (iii) ⇒ 3 + 6α = – 12a
Hence, λ = 0 1 − 2α 20a
∴ =
the equation of the directrix is x + y = 0 3 + 6α − 12a
Using focus-directrix property, the equation or –3(1 – 2α) = 5 (3 + 6α)
of the parabola is
R

or –3 + 6α = 15 + 30α
2
⎛ x+y ⎞ or 24α = – 18
(x –1)2+ (y – = ⎜
1)2 ⎟
3
⎜ 2 2 ⎟ ∴α=–
⎝ 1 + 1 ⎠
CA

4
or 2{(x – 1)2 + (y – 1)2} = (x + y)2
3
or 2(x2 + y2 – 2x – 2y + 2) = x2 + y2 + 2xy ∴ (v) ⇒ 1 + = 20a
2
or x2 + y2 – 2xy – 4x – 4y + 4 = 0
or (x – y)2 = 4 (x + y – 1) 5 1
∴a= =
The latus rectum 40 8
2
⎛1 ⎞ ⎛1
2
⎞
2 ⎛ 3⎞ 1
= 4 × VS = 4 ⎜ − 1 ⎟ + ⎜ − 1 ⎟ ∴ (ii) ⇒ ⎜ − ⎟ = 4 . (4 – β)
2 2 ⎝ 4 ⎠ 8
⎝ ⎠ ⎝ ⎠
1 9 1 9
=4× =2 2 Ans. or = (4 – β), i.e., 4 – β =
16 2 8
2

_________________________________________________________________________ Parabola | 169

 9 23 Ex.4 Prove that two straight lines, one a tangent
or β = 4 – =
8 8 to the parabola y2 = 4a (x + a) and the other
∴ from (i), the equation of the parabola is to the parabola y2 = 4a' (x + a'), which are
2 at right angles to one another, meet on the
⎛ 3⎞ 1 ⎛ 23 ⎞
⎜ x + ⎟ = 4. . ⎜ y − ⎟ straight line x + a + a' = 0. Show also that
⎝ 4⎠ 8 ⎝ 8 ⎠
this straight line is common chord of the
3 9 1 23 two parabolas.
or x2 + x+ = y–
2 16 2 16 Sol. Parabolas y2 = 4a(x + a) ...(i)
3 1 y2 = 4a'(x + a') ...(ii)
or x2 + x – y + 2 = 0 Equation to the tangents of (i) & (ii) are
2 2
∴ 2x2 + 3x – y + 4 = 0 y = m(x + a) + a/m ...(iii)

T
and y = m' (x + a') + a'/m' ...(iv)
1 1
and its latus-rectum = 4a = 4 × = mm' = –1 ...(v)
8 2
Point of intersection of (iii) and (iv)

IN
Ans. a a'
(m – m')x + am – m'a' + − =0
m m'
Ex.3 For what real values of a, the point (–2a, a + 1)
( am'−a' m)
will be an interior point of the smaller ⇒ (m – m' )x + am – m'a' + =0
−1
region bounded by the circle x2 + y2 = 4 and
⇒ (m – m' )x + a (m – m' ) + a' (m – m' ) = 0

PO
the parabola y2 = 4x.
⇒ x + a + a' = 0
Sol. The point P (–2a, a + 1) will be an interior
To get common chord subtract (ii) from (i),
point of both the circle x2 + y2 – 4 = 0 and
We get, x + a + a' = 0
the parabola y2 – 4x = 0.
Hence Proved. Ans.
∴ (–2a)2 + (a + 1)2 – 4 < 0,
i.e., 5a2 + 2a – 3 < 0 ...(i) Ex.5 Find the centre and radius of the smaller
y2 = 4x of the two circles that touch the parabola
⎛6 8⎞
R
P 75y2 = 64 (5x – 3) at ⎜ , ⎟ and the x-axis.
⎝5 5⎠
Sol. The parabola can be written as
75y2 = 32 (10x – 6).
EE

x2 + y2 = 4 Tangent at (x1, y1) is

75yy1 = 32[5 (x + x1) – 6]
and (a + 1)2 – 4(–2a) < 0,
4x–3y = 0
i.e., a2 + 10a + 1 < 0 ...(ii) C2 –r
The required values of a will satisfy both (i) 90º
and (ii). P C1
R

From (i), (5a – 3)(a + 1) < 0

3
∴ by sign-scheme we get, – 1 < a <
5
CA

...(iii) 8 ⎡ ⎛ 6⎞ ⎤
75y. = 32 ⎢5⎜ x + ⎟ − 6⎥
Solving (ii), the corresponding equation is 5 ⎣ ⎝ 5⎠ ⎦
a2 + 10a + 1 = 0 or 4x – 3y = 0
− 10 ± 100 − 4 Above is common tangent to parabola and
or a = =–5±2 6
2 circle at (6/5, 8/5). The centre of the circle
∴ by sign-scheme will be on normal at a distance ± r from the
point of contact. Slope of tangent is 4/3 and
–5 – 2 6 < a < –5 + 2 6 ...(iv)
hence of normal will be – 3/4 = tanθ
The set of values of a satisfying (iii) and
4 3
(iv) is ∴ cosθ = – , sin θ =
5 5
–1<a<–5+2 6. Ans.

170 | Parabola ________________________________________________________________________

 Hence the coordinates of centres are points ∴ R = {at1t2, a(t1 + t2)}
C1 and C2 on the normal through P(6/5, 8/5) ∴ the area of Δ PQR
and at a distance r or –r from P. 2
at1 2at1 1
x − 6/5 y − 8/5 1 2
= = r, – r = at2 2at2 1
− 4/5 3/5 2
at1t2 a(t1 + t2 ) 1
⎛ − 4r + 6 3r + 8 ⎞
∴ C1 = ⎜ , ⎟, 2
⎝ 5 5 ⎠ t1 2t1 1
1 2 2
= a t2 2t2 1
⎛ 4r + 6 − 3r + 8 ⎞ 2
C2 = ⎜ , ⎟ t1t2 t1 + t2 1
⎝ 5 5 ⎠

T
Since the circle touches x-axis, therefore 2
t1 2t1 1
y co-ordinate of centre is radius 1 2
= a (t2 + t1 )(t2 − t1 ) 2(t2 − t1 ) 0
2

IN
3r + 8 t1 (t2 − t1 ) t2 − t1 0
∴ =r ∴ r = 4 for C1
5
R2 → R2 – R1, R3 → R3 – R1
−3r + 8
or = r ∴ r = 1 for C2 1 2 t +t 2
5 = a (t2 – t1)2 2 1
2 t1 1
Hence C1 = (–2, 4) and C2 = (2, 1)

PO
respectively. 1 2
= a (t2 − t1 )3 Ans.
For smaller circle r = 1 and C2 = (2, 1). 2
Ans.
Ex.7 If the tangents at two points of a parabola
Ex.6 The tangents to the parabola y2 = 4ax at are at right angles then they intersect at a
P(t1) and Q(t2) intersect at R. Prove that point on the directrix and the chord joining
the area of the triangle PQR is the two points passes through the focus.
R
Prove this.
1 2
a (t2 – t1)3. Sol. Let the parabola be y2 = 4ax whose focus is
2
(a, 0) and the equation of directrix is x + a = 0.
Sol. The tangent to the parabola y2 = 4ax at
EE

2 2
2 2 Let P ( at1 , 2at1 ) and Q ( at2 , 2at2 ) be two
P ( at1 , 2at1 ) is yt1 = x + at1 ...(i)
points where the tangents are at right
and that at Q( at22 , 2at2) is
angles.
yt2 = x + at22 ...(ii) The tangent to the parabola y2 = 4ax at
R

Solving (i) and (ii) we get the coordinates of R 2 2

P ( at1 , 2at1 ) is yt1 = x + at1 ...(i)
2 2
(i) – (ii) ⇒ y(t1 – t2) = a (t1 − t2 ) and the tangent to the parabola y2 = 4ax at
2 2
Q( at2 , 2at2) is yt2 = x + at2 ...(ii)
CA

P(at12, 2at1)
R Q (i) and (ii) are at right angles,
1 1
∴ . =–1 ...(iii)
Q(at22, 2at2) t1 t2
Also, solving (i) and (ii), the point of
or y = a (t1 + t2) {Q t1 ≠ t2} intersection R of the tangents is
From t2×(i) – t1×(ii), {at1t2, a(t1 + t2)}, i.e., {–a, a(t1 + t2)},
2 2
0 = x(t2 – t1) + a (t1 t2 − t2 t1 ) using (iii)
or 0 = x (t2 – t1) + at1t2 (t1 – t2) But the point R{–a, a(t1 + t2)} satisfies
∴ x = at1t2 {Q t1 ≠ t2} x + a = 0 which is the directrix.

_________________________________________________________________________ Parabola | 171

 ∴ the tangents at right angles meet on the respect to the parabola y2 = 4bx is the
directrix. parabola ay2 = 4b2x.
Also, the equation of the chord PQ is Sol. Any tangent to the parabola
2at1 − 2at2 2 y2 = 4ax is ty = x + at2 ...(i)
y – 2at1 = 2 2
.( x − at1 )
at1 − at2 Let (α, β) be the pole of (i) with respect to
2 2
the parabola y2 = 4bx.
or y – 2at1 = .( x − at1 )
t1 + t2 Then (i) is the polar of (α, β) with respect to
This equation is satisfied by (a, 0) because y2 = 4bx
by putting x = a, y = 0 we have ∴ (i) and yβ = 2b (x + α) are identical.
2 So, comparing these,

T
2
–2at1 = .( a − at1 )
t1 + t2 t 1 at 2
= = ;
or –t1 (t1 + t2) = 1– t1
2 β 2b 2bα

IN
or t1t2 = – 1, which is true from (iii) β 2 α
∴t= , t =
Hence, the chord joining the points passes 2b a
through the focus. Ans. 2
⎛ β ⎞ α
∴ ⎜ ⎟ =
⎝ 2b ⎠ a

PO
Ex.8 Show that the locus of the middle points of
normal chords of the parabola y2 = 4ax is or aβ2 = 4b2α
y4 – 2a(x – 2a)y2 + 8a4 = 0. ∴ The equation of the required locus of the
Sol. If (x1, y1) is the middle point of the chord of poles is ay2 = 4b2x. Ans.
the parabola then T = S1
yy1 – 2a(x + x1) = y12 – 4ax1 Ex.10 Prove that the length of the intercepts on
⇒ 2ax – yy1 + y12 – 2ax1 = 0 ...(i) the normal at the point P (at2, 2at) of the
parabola y2 = 4ax made by the circle
R
Equation of normal of the parabola at 't' is
y = – tx + 2at + at3 described on the line joining the focus and
⇒ tx + y – 2at – at3 = 0 ...(ii) P as diameter is a 1 + t 2
EE

Equation (i) and (ii) are identical then

Sol. The eqn. of the normal PG at P (at2, 2at) to
2
2a − y1 y − 2ax1 y2 = 4ax is y + xt = 2at + at3 ...(i)
= = 1
t 1 − 2at − at 3 Let the circle described on the line PS
from first two relation where S is the focus (a, 0) as diameter cut
2a PG at A. Join SA.
t=– ...(iii)
R

y1 Then ∠SAP = 90º

from first & last relation.
2 ⇒ PA = PS2 − SA 2 ...(ii)
2a(–2a – at2) = y1 – 2ax1 ...(iv)
CA

Substituting the value of t from (iii) in (iv) Now, PS = ( at 2 − a ) 2 + (2at − 0) 2

then
= a t 4 − 2t 2 + 1 + 4t 2
⎛ 4a 3 ⎞
2a⎜ − 2a − 2 ⎟ = y12 − 2ax1 = a (t2 + 1)
⎜ y1 ⎟
⎝ ⎠ Also, SA = length of perpendicular from S
4 2
⇒ y1 –2a (x1 – 2a) y1 + 8a4 = 0 to (i)
Hence locus of the middle point (x1, y1) is
y4 – 2a (x – 2a)y2 + 8a4 = 0 Ans.
Ex.9 Show that the locus of the poles of the
tangents to the parabola y2 = 4ax with

172 | Parabola ________________________________________________________________________

 y P(at2,2at) Ex.12 P, Q are the points t1, t2 on the parabola
y2 = 4ax. The normals at P, Q meet on the
A
x parabola. Show that t1t2 = 2 and that the
O S (a, 0)
G middle point of PQ lies on the parabola
y2 = 2a(x + 2a).
− at + 2at + at 3 Sol. The eqn. of parabola be y2 = 4ax
= 2
t2 + 1 Eqn. of normal at P ( at1 , 2at1 ) is
3
= at t2 + 1 y = – t1x + 2at1 + at1 ...(i)
So from eqn. (ii), 2
Eqn. of normal at Q ( at2 , 2at2 ) is

T
PA = a 2 (t 2 + 1)2 − a 2t 2 (t 2 + 1) y = – t2x + 2at2 + at2
3
...(ii)

IN
2
= a t +1 t +1 − t 2 2 Eqn. (i) – eqn. (ii) gives
2 2
(t1 – t2)x = 2a(t1 – t2) + a(t1 – t2) (t1 + t2 + t1t2 )
= a 1 + t2 Ans.
2 2
⇒ x = 2a + a (t1 + t2 + t1t2 )
Ex.11 The tangent and normal at any point of a Putting this value in eqn. (i), we get

PO
parabola are the bisectors of the angles y = – at1t2 (t1 + t2)
between the focal chord through the point So, co-ordinates of L are
and the perpendicular on the directrix from 2 2
[2a + a (t1 + t2 + t1t2 ) , – at1t2 (t1 + t2)]
the point.
Since L lies on y2 = 4ax,
Sol. Let P(at2, 2at) be any point of the parabola
y
y2 = 4ax, Q(at22,2at2)
Equation of tangent at 'P' is ty = x + at2 P(at12,2at1)
R
Slope of tangent = 1/t, x
O
2t
Slope of SP = 2
t −1 L
EE

2 2
Normal
⇒ a 2t1 t2 (t1 + t2)2
y
Tangent 2 2
= 4a2[2 + (t1 + t2 + t1t2 ) ]
M θ P 2 2
⇒ t1 t2 (t1 + t2)2 = 4 [(t1 + t2)2 – (t1t2 – 2)]
θ x
R

O 2 2
T S(a, 0) ⇒ (t1 t2 − 4 ) (t1 + t2)2 + 4 (t1t2 – 2) = 0
⇒ (t1t2 – 2) [(t1t2 + 2)(t1 + t2)2 + 4] = 0
Directrix ⇒ either t1t2 = 2 or (t1t2 + 2) (t1 + t2)2 + 4 = 0
CA

⎛ 2t 1⎞ Now, mid-point of PQ is
⎜ 2 − ⎟
t −1 t ⎠ 4a
∴ tan ∠ SPT = ⎝
2
2 2
at1 + 2
2t 1 at1 + at2 t1
1+ 2 . h= = [Q t1t2 = 2]
t −1 t 2 2
1 4
at1 + 4a
4
a(t1 + 4 )
= = tan ∠PTS = =
t 2t1
2
2t1
2

∴ ∠SPT = ∠PTS
2a(t1 + t2 )
Hence ∠TPM = ∠SPT Ans. ⇒k=
2

_________________________________________________________________________ Parabola | 173

 ⎛ 2⎞ Ex.14 If the normals at P, Q, R of the parabola
= a ⎜⎜ t1 + ⎟⎟ [Q t1t2 = 2]
⎝ t1 ⎠ y2 = 4ax meet in O and S be its focus. Prove
that SP . SQ . SR = α . (SO)2.[Property]
⎛t2 + 2⎞
= a⎜ 1 ⎟ Sol. Normal at t is y + tx = 2at + at3
⎜ t1 ⎟
⎝ ⎠ it is passes through O (h, k) then
Putting these values in y2 = 2a (x + 2a), we at3 + (2a – h) t – k = 0
get Y
2
a 2 (t1 + 2) 2 ⎡ a(t14 + 4 ) ⎤ Q
2
= 2a ⎢ 2
+ 2a ⎥ M
t1 ⎢⎣ 2t1 ⎥⎦ P

T
2 X′ S(a, 0) X
a 2 (t1 + 2) 2 a2 4 2 A
⇒ 2
= 2
[t1 +4 + 4t1 ] O(h, k)
t1 t1

IN
R
a2 Y′
= 2
(t12 + 2) 2 If the co-ordinates of P, Q, R are
t1
2 2 2
Thus L.H.S. = R.H.S. ( at1 ,2at1 ) , ( at2 ,2at2 ) , ( at3 ,2at3 ) then
Hence mid-point of PQ lies on ∑t1 = 0

PO
y2 = 4a (x + 2a). Ans. 2a − h
Σ t1t2 =
a
Ex.13 The normal at any point 'P' meets the axis t1t2t3 = k/a
in G and the tangent at the vertex in G' ; if 2
then SP = PM = a + at1
A be the vertex and the rectangle AGQG'
2
be completed, prove that the equation to Similarly SQ = a + at2 and
the locus of Q is x3 = 2ax2 + ay2. 2
R
SR = a + at3
Sol. Normal at P(am2, –2am) is 2 2 2
∴ SP. SQ. SR = a3 (1 + t1 ) (1 + t2 ) (1 + t3 )
y = mx – 2am – am3
2 2 2 2 2 2
= a3 {1 + ∑ t1 + ∑ t1 t2 + t1 t2 t3 }
EE

Q(h,k)
G´ = a3{(1 + (Σt1)2 – 2Σt1t2 + (Σt1t2)2
P
x – 2t1t2t3Σt1 + t12 t22 t32 }
A G
⎧⎪ 2(2a − h ) (2a − h )2 k 2 ⎪⎫
= a3 ⎨1 + 0 − + − 0 + ⎬
⎪⎩ a a2 a 2 ⎪⎭
R

It meet the axis of parabola (x-axis) at

y = 0, x = 2a + am2 = a(a2 – 2a(2a – h) + (2a – h)2 + k2}
⇒ G ≡ (2a + am2, 0) = a{h2 + a2 – 2ah + k2}
CA

and the tangent at the vertex (y-axis) at = a{(h – a)2 + k2}

x = 0, y = –2am – am2 Hence, SP . SQ . SR. = a(SO)2 Ans.
⇒ G' ≡(0, –2am – am3)
∴ h = 2a + am2 ...(i) Ex.15 A variable chord PQ of the parabola y = x2
k = –2am – am3 ...(ii) subtends a right angle at the vertex. Find
Eliminating m from (i) & (ii), we get the locus of points of intersection of the
h3 = 2ah2 + ak2 normals at P and Q.
Now taking the locus of Q(h, k) Sol. The vertex V of the parabola is (0, 0) and
we get x3 = 2ax2 + ay2. Ans. any point on y = x2 has the coordinates
(t, t2)

174 | Parabola ________________________________________________________________________

 So let us take P = (t1, t12 ), Q = (t2, t22 ) and ⎧⎪⎛ 1 ⎞ 2 ⎫⎪
= 2 ⎨⎜ x ⎟ + 1⎬ + 1, using (i) and (vi)
∠PVQ = 90º ⎪⎩⎝ 2 ⎠ ⎪⎭
P(t1,t12) ∴ The equation of the required locus is
M x2
V(0, 0) 2y = +3
2
Q(t2,t22) or x2 = 2(2y – 3), which is a parabola. Ans.

2
t1 − 0 Ex.16 Prove that three normals can be drawn
As 'm' of VP = = t1
t1 − 0
from the point (c, 0) to the parabola y2 = x if

T
2
t2 −0 1
and 'm' of VQ = = t2 c > and then one of the normals is
t2 − 0 2

IN
QVP ⊥ VQ ∴t1 t2 = – 1 ...(i) always the axis of the parabola. Also find c
The equation of the normal to a curve at for which the other two normals will be
(x1, y1) is perpendicular to each other.
Sol. Let (t2, t) be a foot of one of the normals to
−1 .( x − x1 )
y – y1 = the parabola y2 = x from the point (c, 0).

PO
⎛ dy ⎞
⎜ ⎟ Now, the equation of the normals to y2 = x
⎝ dx ⎠( x 1
, y1 )
at (t2, t) is
dy
Here, y = x2 ; ∴ = 2x −1
dx y–t= .( x − t 2 )
⎛ dy ⎞
⎜ ⎟
∴ The equation of the normal at P (t1, t12 ) ⎝ dx ⎠(t 2 ,t )
is −1
or y – t = (x − t 2 )
R
2 −1 1
y – t1 = ( x − t1 )
2t1 2t

or 2t1y + x = 2 t1 + t1
3
...(ii) ⎧ 2 dy dy 1 ⎫
⎨Q y = x ⇒ 2 y = 1; ∴ = ⎬
EE

⎩ dx dx 2 y ⎭
Similarly, the equation of the normal at
2 or y – t = – 2t(x – t2)
Q (t2 ,t2 ) is
or y + 2tx = t + 2t3 ...(i)
3
2t2y + x = 2 t2 + t2 ...(iii) It passes through (c, 0) if 0 + 2tc = t + 2t3
Eliminating t1, t2 from (i), (ii) and (iii) we or 2t3 + t (1 – 2c) = 0
R

get the locus of M. or t{2t2 – (2c – 1)} = 0

3
(ii) – (iii) ⇒ 2y(t1 – t2) = 2 (t1 − t2 ) + (t1 – t2)
3 2c − 1
∴ t = 0, ±
2
CA

2 2
or 2y = 2 (t1 + t1t2 + t2 ) + 1 ...(iv)
Three normals can be drawn if t has three
Also, t2 × (ii) – t1 × (iii) real distinct values.
3 3
⇒ (t2 – t1)x = (2t1 + t1 ) t2 – (2t2 + t 2 ) t1 1
So, 2c – 1 > 0, i.e., c >
= 2t1t2 (t1 − t2 )
2 2 2
The foot of one of the normal is (t2, t) where
or x = – 2t1t2 (t1 + t2) ...(v)
t = 0,
From (i) and (v), x = 2 (t1 + t2)
i.e., the foot is (0, 0).
1
or t1 + t2 = x ...(vi) By (i), the corresponding normal is y = 0,
2
From (iv), 2y = 2{(t1 + t2)2 – t1t2} + 1
_________________________________________________________________________ Parabola | 175
 i.e., the x-axis which is the axis of the Ex.18 The general equation to a system of
parabola. 25
parallel chords of the parabola y2 = x is
2c − 1 7
For the other two normals t = ±
2 4x – y + k = 0. What is the equation to the
By (i), 'm' of a normal = – 2t corresponding diameter ?
∴'m' of the other two normals are Sol. Let PQ be a chord of the system whose
equation is 4x – y + k = 0 ...(i)
2c − 1 2c − 1
–2 , 2 where k is a parameter.
2 2
These are perpendicular if Let M (α, β) be the middle point of PQ. The
locus of M is the required diameter.

T
2c − 1 2c − 1
–2 ×2 =–1 25
2 2 The equation of the parabola is y2 = x
7

IN
or –2(2c – 1) = – 1
...(ii)
3
or –4c = – 3 ; c = Ans. 2
7y
4 Solving (i) and (ii), 4 × –y+k=0
25
Ex.17 Let (xr, yr); r = 1, 2, 3, 4 be the points of or 28y2 – 25y + 25k = 0.

PO
intersection of the parabola y2 = 4ax and Let its roots be y1, y2
the circle x2 + y2 + 2gx + 2fy + c = 0. Prove y1 + y2 25 25
Then β = = =
that y1 + y2 + y3 + y4 = 0. 2 2.28 56
Sol. Let x2 + y2 + 2gx + 2fy + c = 0 ...(i) ∴ The equation of the locus of M(α, β) is
y2 = 4ax ...(ii) 25
y= Ans.
Solving (i) and (ii) we get the coordinates of 56
points of intersection.
R
y2
From (ii), x = . Putting in (i),
4a
EE

2
⎛ y2 ⎞ ⎛ 2⎞
⎜ ⎟ + y2 + 2g ⎜ y ⎟ +2fy + c = 0
⎜ 4a ⎟ ⎜ 4a ⎟
⎝ ⎠ ⎝ ⎠
1 ⎛ g ⎞ 2
or y4 + ⎜1 +
2
⎟ y + 2fy + c = 0
(4a ) ⎝ 2a ⎠
R

It has four roots.

Its roots are y1, y2, y3 and y4
coefficient of y 3
Now, sum of roots = –
CA

coefficient of y 4
0
y1 + y2 + y3 + y4 = – =0 Ans.
1 /( 4a )2

176 | Parabola ________________________________________________________________________

 EXERCISE (Level-1)
Question Q.7 The locus of the point of intersection of
based on Different forms of parabola
perpendicular tangent to the parabola
Q.1 The equation of the parabola whose focus is x2 – 8x + 2y + 2 = 0 is-
(1, 1) and tangent at the vertex is x + y = 1 (A) 2y – 15 = 0 (B) 2y + 15 = 0
is (C) 2x + 9 = 0 (D) None of these
(A) x2 + y2 – 2xy – 4x – 4y + 4 = 0
Question
(B) x2 + y2 – 2xy + 4x + 4y + 4 = 0 based on Parametric form
(C) + – 2xy – 4x – 4y – 4 = 0
x2 y2
Q.8 Write the parametric equations of the
(D) None of these
parabola (x + 1)2 = 4 (y – 1)
(A) x = 2t – 1, y = t2 + 1
Q.2 The equation of the parabola which passes (B) x = 2t + 3, y = t2 + 2
through the point (4, 3) and having origin (C) x = 2t – 1, y = t2 – 1
as its vertex and x-axis as its axis will be (D) None of these
(A) 9y2 = 4x (B) 9y2 + 4x = 0
(C) 4y2 + 9x = 0 (D) 4y2 – 9x = 0 Q.9 Which of the following are not parametric
coordinates of any point on the parabola
Q.3 If the vertex = (2, 0) and the extremities of y2 = 4ax
the latus rectum are (3, 2) and (3, –2) then (A) (at2, 2at) (B) (a, 2a)
the equation of the parabola is ⎛ a 2a ⎞
(C) ⎜ 2 , ⎟ (D) (am2, –2am)
(A) y2 = 2x – 4 (B) x2 = 4y – 8 ⎝m m⎠
(C) y2 = 4x – 8 (D) None of these
Question
based on Focal chord
Q.4 The equation of the parabola whose vertex
and focus are on the positive side of the Q.10 The other extremity of the focal chord of
x-axis at distances a and b respectively the parabola y2 = 8x which is drawn at the
from the origin is ⎛1 ⎞
point ⎜ , 2 ⎟ is
(A) y2 = 4(b – a) (x – a) ⎝2 ⎠
(B) y2 = 4(a – b) (x – b) (A) (2, –4) (B) (2, 4)
(C) x2 = 4(b – a) (y – a) (C) (8, –8) (D) (8, 8)
(D) None of these
Q.11 Length of focal chord drawn at point (8, 8)
Q.5 The length of the latus-rectum of the of parabola y2 = 8x is
parabola x2 – 4x – 8y + 12 = 0 is- (A) 25 (B) 18
(A) 4 (B) 6 25 25
(C) 8 (D) 10 (C) (D)
4 2

Q.6 For the parabola y2 + 8x – 12y + 20 = 0, Q.12 A circle described on any focal chord of
which of the following is not correct- parabola y2 = 4ax as its diameter touches
(A) vertex (2, 6) (A) Axis of Parabola
(B) focus (0, 6) (B) directrix of Parabola
(C) length of the latus rectum = 4 (C) Tangent drawn at vertex
(D) axis is y = 6 (D) Latus Rectum
Question Question
based on Tangent of parabola based on Normal of parabola

Q.13 The equation to the line touching both the Q.20 The equations of the normals at the ends of
parabolas y2 = 4x and x2 = – 32y is the latus rectum of the parabola y2 = 4ax are
(A) x + 2y + 4 = 0 (B) 2x + y – 4 = 0 given by
(C) x – 2y – 4 = 0 (D) x – 2y + 4 = 0 (A) x2 – y2 – 6ax + 9a2 = 0
(B) x2 – y2 – 6ax – 6ay + 9a2 = 0
Q.14 The equation of common tangent to the (C) x2 – y2 – 6ay + 9a2 = 0
circle x2 + y2 = 2a2 and parabola y2 = 8ax is- (D) None of these
(A) y = x + a (B) y = ±x ± 2a
(C) y = –x + a (D) y = –x + 2a Q.21 If a normal to the parabola y2 = 8x makes
45º angle with positive direction of x-axis
Q.15 The equation of the common tangent of the
then its foot of the normal will be
parabolas x2 = 108y and y2 = 32x, is-
(A) (2, 4) (B) (2, –4)
(A) 2x + 3y = 36 (B) 2x + 3y + 36 = 0
(C) (8, 8) (D) (8, –8)
(C) 3x + 2y = 36 (D) 3x + 2y + 36 = 0

Q.22 If the normal at the point (1, 2) on the

Q.16 The equation of the tangents to the
parabola y2 = 4x meets the parabola again
parabola y2 = 8x inclined at 45º to the
at the point (t2, 2t), then t is equal to
x-axis and also the points of contact will be
(A) 1 (B) –1
(A) x – y – 2 = 0, (2, 4)
(C) 3 (D) – 3
(B) x – y + 2 = 0, (2, 4)
(C) x – y + 2 = 0, (2, 3)
Q.23 If P (–3, 2) is one end of the focal chord PQ
(D) None of these
of the parabola y2 + 4x + 4y = 0, then the
slope of the normal at Q is
Q.17 A tangent to the parabola y2 = 8x makes an
1
angle of 45º with the straight line y = 3x + 5. (A) – (B) 2
2
The equation of the tangent and its point of
1
contact are (C) (D) –2
2
⎛1 ⎞
(A) 2x + y –1 = 0 at ⎜ , − 2 ⎟ ,
⎝ 2 ⎠ Q.24 The equation of the normal having slope m
x – 2y – 8 = 0 at (8, 8) of the parabola y2 = x + a is
(B) 2x + y + 1 = 0 at (1, – 2), (A) y = mx – 2am – am3
x + 2y + 8 = 0 at (8, 8) (B) y = mx – am – am3
⎛1 ⎞ (C) 4y = 4mx + 4am – 2m – m3
(C) 2x + y + 1 = 0 at ⎜ , − 2 ⎟ ,
⎝2 ⎠ (D) 4y = 4mx + 2am – am3
x – 2y + 8 = 0 at (8, 8)
(D) None of these Q.25 The slopes of the three normals to the
parabola y2 = 8x which pass through
Q.18 The equation of the line touching both the
parabolas y2 = x and x2 = y is (18, 12) are
(A) 4x + 4y + 1 = 0 (B) 4x + 4y – 1 = 0 (A) 1, 2, 3 (B) 1, –2, 3
(C) x + y + 1 = 0 (D) None of these (C) 1, 2, –3 (D) –1, –2, 3

Q.19 If the chord y = mx + c subtends a right

Q.26 Number of normals can be drawn from
angle at the vertex of the parabola
point (1, 2) on the parabola y2 = 12x is
y2 = 4ax, then the value of c is-
(A) 3 (B) 1
(A) –4am (B) 4am
(C) –2am (D) 2am (C) 2 (D) None of these
Question Q.31 The circle x2 + y2 + 2λx = 0, λ ∈ R, touches
based on Tangents from external point
the parabola y2 = 4x externally. Then
Q.27 The tangents from the origin to the (A) λ > 0 (B) λ < 0
parabola y2 + 4 = 4x are inclined at (C) λ > 1 (D) None of these
π π
(A) (B) Q.32 The equation of the director circle of the
6 4
parabola x2 = 4ay is
π π
(C) (D) (A) x2 + y2 = a2 (B) x2 + y2 = 2a2
3 2
(C) x + a = 0 (D) y + a = 0

Q.28 The equation of the chord of contact of

T
Question
tangents drawn from the point (2, 3) to the based on Chord with mid point
parabola y2 + x = 0 is Q.33 If (a, b) be the mid point of a chord of the

IN
(A) 3y + x = 2 (B) 6y – x = 2 parabola y2 = 4x passing through its vertex
(C) 6y + x + 2 = 0 (D) 3y – x = 2 then
(A) a = 2b (B) 2a = b
Question
based on Chord of contact (C) a2 = 2b (D) 2a = b2

PO
Q.29 Line x + y = 2 meets parabola y2 = 8x at Q.34 The mid-point of the line joining the
point P and Q. Point of intersection of common points of the line 2x – 3y + 8 = 0
tangents drawn at P and Q is and y2 = 8x is
(A) (– 2, – 4) (B) (– 1, – 4) (A) (3, 2) (B) (5, 6)
(C) (– 2, – 3) (D) (– 3, – 2) (C) (4, –1) (D) (2, –3)

Q.30 The chord of contact of the tangents to a Q.35 If the tangent at the point P(2, 4) to the
parabola drawn from any point on its parabola y2 = 8x meets the parabola y2 = 8x + 5
directrix passes through at Q and R, then the mid-point of QR is
(A) one extremity of LR (A) (2, 4) (B) (4, 2)
(B) focus (C) (7, 9) (D) None of these
(C) vertex
(D) None of these

_________________________________________________________________________ Parabola | 179

 EXERCISE (Level-2)
Single correct answer type questions Q.7 If a tangent line at a point P on a parabola
makes angle α with its focal distance, then
Q.1 A tangent to the parabola y2 = 4ax at P (p, q) angle between the tangent and axis of the
is perpendicular to the tangent at the other parabola is-
point Q, then coordinates of Q are- (A) α (B) α/2
⎛ a2 4a 2 ⎞⎟ ⎛ a2 4a 2 ⎞ (C) 2α (D) 90º
(A) ⎜ ,− (B) ⎜ − ,− ⎟
⎜ p q ⎟⎠ ⎜ p q ⎟
⎝ ⎝ ⎠ Q.8 If a focal chord of parabola y2 = 4ax makes
⎛ a 4a ⎞
2 2 ⎛ a 4a ⎞
2 2

T
an angle θ with its axis, then the length of
(C) ⎜ − , ⎟ (D) ⎜ , ⎟
⎜ p q ⎟ ⎜ p q ⎟ perpendicular from vertex to this chord is-
⎝ ⎠ ⎝ ⎠
(A) a tan θ (B) a cos θ

IN
(C) a sin θ (D) a sec θ
Q.2 The coordinates of a point on the parabola
y2 = 8x whose focal distance is 4 is-
Q.9 An equilateral triangle is inscribed in the
(A) (2, 4) (B) (4, 2) parabola y2 = 4x whose vertex is at the
(C) (2, –9) (D) (4, –2) vertex of the parabola. The area of this

PO
triangle is-
Q.3 PQ is a double ordinate of y2 = 4ax. The (A) 48 3 (B) 16 3
locus of its point of trisection is
(C) 64 3 (D) 8 3
(A) y2 = 2ax (B) 3y2 = 4ax
(C) 9y2 = 4ax (D) 9y2 = 2ax
Q.10 A circle with centre at the focus of the
parabola y2 = 4px touches the directrix.
Q.4 Which one of the following represented
Then a point of intersection of the circle
parametrically, represents equation of a
R
and the parabola is-
parabola-
(A) (–p, 2p) (B) (p, –2p)
(A) x = 3 cos t; y = 4 sin t
(C) (p, ±2p) (D) (–p, –2p)
t
EE

(B) x2 – 2 = –2 cos t; y = 4 cos2

2 Q.11 L(2, 4) and L′(2, –4) are the ends of the
(C) x = tan t; y = sec t latus- rectum of a parabola. P is a point on
t t the directrix. Then the area of ΔPLL′ =
(D) x = 1 − sin t ; y = sin + cos
2 2 (A) 16 (B) 8
(C) 4 (D) 1
R

Q.5 If (x1, y1) and (x2, y2) and ends of a focal

chord of the parabola y2 = 4ax, then square Q.12 The equation of the locus of a point which
of G.M. of x1 and x2 is- moves so as to be at equal distances from
CA

(A) –4a2 (B) 4a2 the point (a, 0) and the y-axis is
(C) a 2 (D) –a2 (A) y2 – 2ax + a2 = 0 (B) y2 + 2ax + a2 = 0
(C) x2 – 2ay + a2 = 0 (D) x2 + 2ay + a2 = 0
Q.6 The length of L.R. of the parabola
Q.13 If the line x – 1 = 0 is the directrix of the
gx 2
y = x tanα – is- parabola y2 – kx + 8 = 0, then one of the
2u 2 cos2 α values of k is-
2u 2 cos 2 α u 2 sin 2 2α 1
(A) (B) (A) (B) 8
g g 8
u 2 cos2 2α 1
(C) (D) None of these (C) 4 (D)
g 4

180 | Parabola ________________________________________________________________________

Q.14 Equation of the directrix of the parabola Q.21 The equation of directrix of a parabola is
whose focus is (0, 0) and the tangent at the 3x + 4y + 15 = 0 and equation of tangent at
vertex is x – y + 1 = 0 is vertex is 3x + 4y – 5 = 0.Then the length of
latus rectum is equal to
(A) x – y = 0 (B) x – y – 1 = 0
(A) 15 (B) 14
(C) x – y + 2 = 0 (D) x + y – 1 = 0
(C) 13 (D) 16

Q.15 The point on y2 = 4ax nearest to the focus Q.22 Length of the tangent drawn from an end
has its abscissae equal to of the latus rectum of the parabola y2 = 4ax
(A) –a (B) a to the circle of radius a touching externally
(C) a/2 (D) 0 the parabola at vertex is equal to-
(A) 3a (B) 2a

T
Q.16 The angle subtended by the double (C) 7a (D) 3a
ordinate of length 2a of the parabola

IN
y2 = ax, at the vertex is equals to- Q.23 If M is the foot of perpendicular from a
π π point P of a parabola y2 = 4ax to its
(A) (B)
4 3 directrix and SPM is an equilateral
π triangle, where S is the focus. Then SP
(C) (D) None of these equal to-
2

PO
(A) a (B) 2a
(C) 3a (D) 4a
Q.17 The equation of a tangent to the parabola
y2 = 12x is y = x + 3. The point on this line Q.24 If the point P(2, –2) is the one end of the
from which other tangent to the parabola is focal chord PQ of the parabola y2 = 2x then
perpendicular to the given tangent is- slope of the tangent at Q is-
(A) (0, 4) (B) (–3, 3) (A) –2 (B) 2
(C) (–2, 3) (D) None of these 1 1
(C) (D) –
R
2 2
Q.18 If three points E, F, G are taken on the
parabola y2 = 4ax so that their ordinates Q.25 If t1 and t2 be the ends of a focal chord
of the parabola y2 = 4ax, then the
EE

are in G.P., then the tangents at E and G

equation t1x2 + ax + t2 = 0 has
intersect on the- (A) imaginary roots
(A) directrix (B) axis (B) both roots positive
(C) ordinate of F (D) None of these (C) one positive and one negative roots
(D) both roots negative
Q.19 A circle has its centre at the vertex of the
R

parabola x2 = 4y and the circle cuts the Q.26 The length of a focal chord of the parabola
parabola at the ends of its latus rectum. y2 = 4ax at a distance b from the vertex is c.
The equation of the circle is- Then
CA

(A) x2 + y2 = 5 (B) x2 + y2 = 4 (A) 2a2 = bc (B) a3 = b2c

(C) x + y = 1
2 2 (D) None of these (C) ac = b2 (D) b2c = 4a3

Q.20 A triangle ABC of area 5a2 is inscribed in Q.27 The shortest distance between line x + y = 3
the parabola y2 = 4ax such that vertex A and parabola whose directrix is x + y = 1
lies at the vertex of parabola and BC is a and focus at (–1, –1) is
focal chord. Then the length of focal chord is-
3 3
25a (A) (B) + 2
(A) 5a (B) 2 2 2
4
3
5a (C) + 2 (D) None of these
(C) (D) 25a 2 2
4
_________________________________________________________________________ Parabola | 181
Q.28 The point (a, 2a) is an interior point of the Q.35 The point on the line x – y + 2 = 0 from
region bounded by the parabola y2 = 16x which the tangent to the parabola y2 = 8x is
and the double ordinate through the focus. perpendicular to the given line is (a, b),
Then a belongs to the open interval then the line ax + by + c = 0 is
(A) a < 4 (B) 0 < a < 4 (A) parallel to x-axis
(C) 0 < a < 2 (D) a > 4
(B) parallel to y-axis
(C) equally inclined to the axes
Q.29 The vertex of the parabola y2 = 8x is at the
centre of a circle and the parabola cuts the (D) None of these
circle at the ends of its latus rectum. Then
the equation of the circle is Q.36 If two tangents drawn from the point (α, β)

T
(A) x2 + y2 = 4 (B) x2 + y2 = 20 to the parabola y2 = 4x be such that the
(C) x2 + y2 = 80 (D) None of these slope of one tangent is double of the other
then

IN
Q.30 The co-ordinates of the point on the 2 2
parabola y2 = 8x, which is at minimum (A) β = α2 (B) α = β2
9 9
distance from the circle x2 + (y + 6)2 = 1 are
(C) 2α = 9β2 (D) None of these
(A) (2, 4) (B) (–2, 4)
(C) (2, –4) (D) None of these
Q.37 If line 3x + y = 8 meets parabola

PO
(y – 2)2 = 4(x – 1) at A and B, then the point
Q.31 Two parabolas y2 = 4a(x – λ1) & x2 = 4a(y –λ2)
of intersection of tangents drawn at A and
always touch each other, λ1 and λ2 being
B lies on line
variable parameters. Then, their points of
1
contact lie on a (A) x = –1 (B) x = –
(A) Straight line (B) Circle 2
(C) Parabola (D) Hyperbola (C) x = 0 (D) None of these

Q.32 The area of the triangle formed by the Q.38 Let tangent at P(3, 4) to the parabola
R
tangent and the normal to the parabola (y – 3)2 = (x – 2) meets line x = 2 at A and if
y2 = 4ax, both drawn at the same end of the S be the focus of parabola then ∠SAP is
latus rectum, and the axis of the parabola equal to
EE

is π π
(A) (B)
(A) 2 2a 2 (B) 2a2 4 2
(C) 4a2 (D) None of these π
(C) (D) None of these
3
Q.33 If two of the three feet of normals drawn
from a point to the parabola y2 = 4x be Q.39 Angle between the tangents drawn from
R

(1, 2) and (1, –2) then the third foot is (1, 4) to the parabola y2 = 4x is
(A) (2, 2 2 ) (B) (2, – 2 2 ) π π
(A) (B)
(C) (0, 0) (D) None of these 2 3
CA

π π
(C) (D)
Q.34 Let the line lx + my = 1 cut the parabola 6 4
y2 = 4ax in the points A and B. Normals at
A and B meet at point C. Normal from C Q.40 The locus of the mid-point of the line
other than these two meet the parabola at D segment joining the focus to a moving point
then the coordinate of D are on the parabola y2 = 4ax is another
⎛ 4am 4a ⎞ parabola with directrix-
(A) (a, 2a) (B) ⎜ 2 , ⎟
⎝ l l ⎠ a
(A) x = – a (B) x = –
2
⎛ 2am 2 2a ⎞ ⎛ 4am 2 4am ⎞
(C) ⎜ 2 , ⎟ (D) ⎜ 2 , ⎟ a
⎜ l l ⎟⎠ ⎜ l l ⎟⎠ (C) x = 0 (D) x =
⎝ ⎝ 2

182 | Parabola ________________________________________________________________________

 EXERCISE (Level-3)
Q.6 A tangent to the parabola y2 = 4ax is
Part-A : Multiple correct answer type questions
inclined at an angle π/3 with the axis of the
Q.1 A square has one vertex at the vertex of parabola. The point of contact is
the parabola y2 = 4ax and one of the ⎛ a – 2a ⎞
(A) ⎜⎜ , ⎟
⎟ (B) (3a, –2 3 a)
diagonal through this vertex lies along the ⎝3 3 ⎠
axis of the parabola. If the ends of the
⎛ a 2a ⎞
other diagonal lie on the parabola, the (C) (3a, 2 3 a ) (D) ⎜ , ⎟
⎜3 3 ⎟
coordinates of the vertices of the squares ⎝ ⎠

T
are
(A) (4a, 4a) (B) (4a, –4a) Q.7 If a ≠ 0 and the line 2bx + 3cy + 4d = 0

IN
(C) (8a, 0) (D) (4 2 a, 0) passes through the points of intersection of
the parabolas y2 = 4ax and x2 = 4ay, then-
Q.2 Let the equations of a circle and a parabola (A) d2 + (2b + 3c)2 = 0
be x2 + y2 – 4x – 6 = 0 and y2 = 9x (B) d2 + (3b + 2c)2 = 0

PO
respectively. Then (C) d2 + (2b – 3c)2 = 0
(A) (1, –1) is a point on the common chord (D) d2 + (3b – 2c)2 = 0
(B) the equation of the common chord is
y+1=0 Q.8 The normals to the parabola y2 = 4ax from
(C) the length of the common chord is 6 the point (5a, 2a) are
(D) None of these (A) y = x – 3a (B) y = –2x + 12a
(C) y = –3x + 33a (D) y = x + 3a
Q.3 Above x-axis, the equation of the common
R
tangents to the circle (x – 3)2 + y2 = 9 and Q.9 Which of the following is (are) true about
parabola y2 = 4x is- the parabola y2 = 4ax (a > 0) ?
(A) 3 y = 3x + 1 (B) 3 y = – (x + 3) (A) If t1, t2 are end points of a focal chord,
EE

(C) 3 y=x+3 (D) 3 y = – (3x + 1) then t1t2 = –1

(B) Tangent at the end of a focal chord cuts
Q.4 Consider a circle with centre lying on the at right angle at directrix
focus of the parabola y2 = 2px such that it (C) Distance of any point on the parabola
touches the directrix of the parabola. Then
R

from directrix is equal to the sum of 'a'

a point of intersection of the circle and the and abscissa of the point
parabola is (D) End points of latus rectum are (a, 2a),
⎛p ⎞ ⎛p ⎞
CA

(A) ⎜ , p ⎟ (B) ⎜ , − p ⎟ (–a, 2a)

⎝2 ⎠ ⎝2 ⎠
⎛ p ⎞ ⎛ p ⎞ Q.10 The locus of the mid point of the focal radii
(C) ⎜ − , p ⎟ (D) ⎜ − , − p ⎟
⎝ 2 ⎠ ⎝ 2 ⎠ of a variable point moving on the parabola,
y2 = 8x is a parabola whose
Q.5 The slope of a tangent to the parabola y2 = 9x (A) Latus rectum is half the latus rectum of
which passes through the point (4, 10) is
the original parabola
9 1
(A) (B) (B) Vertex is (1, 0)
4 4
(C) Directrix is y-axis
3 1
(C) (D) (D) Focus has the co-ordinates (2, 0)
4 3
_________________________________________________________________________ Parabola | 183
Q.11 If from the vertex of a parabola y2 = 4x, a Q.15 Assertion (A) : Through (h, h + 1) there
pair of chords be drawn at right angles to cannot be more than one normal to the
one another and with these chords as parabola y2 = 4x, if h < 2.
adjacent sides a rectangle be made, then Reason (R) : The point (h, h + 1) lies
the locus of the further end of the rectangle is outside the parabola for all h ≠ 1.
(A) an equal parabola
(B) a parabola with focus at (9, 0) Q.16 Assertion (A) : The latus rectum of a
(C) a parabola with directrix as x – 7 = 0 parabola is 4 unit, axis is the line 3x + 4y –
(D) a parabola having tangent at its vertex 4 = 0, and the tangent at the vertex is the
line 4x – 3y + 7 = 0 then the equation of
x=8
directrix of the parabola is 4x – 3y + 8 = 0.

T
Reason (R) : If P be any point on the
Q.12 Let L be a normal to the parabola y2 = 4x.
parabola and let PM and PN are
If L passes through the point (9, 6), then L
perpendiculars from P on the axis and
is given by

IN
tangent at the vertex respectively, then
(A) y – x + 3 = 0 (B) y + 3x – 33 = 0 (PM)2 = (latusrectum) (PN)
(C) y + x – 15 = 0 (D) y – 2x + 12 = 0
Part-C : Column Matching type Questions
Part-B : Assertion Reason type Questions

PO
The following questions 13 to 16 consists of Q.17 In column I different equations of
parabolas are given and in column II their
two statements each, printed as Assertion
one of the parameters is given match them.
and Reason. While answering these questions
Column I Column II
you are to choose any one of the following
four responses. (A) Vertex of parabola ⎛5 ⎞
(P) ⎜ , 2 ⎟
(A) If both Assertion and Reason are true x2 – 6x – 4y + 3 = 0 ⎝4 ⎠
and the Reason is correct explanation of (B) Focus of parabola ⎛–5 ⎞
(Q) ⎜ , 2⎟
the Assertion. y2 – 4y – 3x – 2 = 0 ⎝ 4 ⎠
R
(B) If both Assertion and Reason are true but
(C) Mid point of vertex & (R) ⎛ 3⎞
Reason is not correct explanation of the ⎜ 3, – ⎟
Focus of parabola ⎝ 2⎠
Assertion.
EE

(C) If Assertion is true but Reason is false. (y – 2)2 = 2 (x – 1)

(D) Point at which normal (S) (1, 2)
(D) If Assertion is false but Reason is true
drawn on parabola
y2 = 4x, makes equal
Q.13 Assertion (A) : The latus rectum is the
angle with axis
shortest focal chord in a parabola y2 = 4ax.
Reason (R) : As the length of focal chord
R

2 Q.18 Match the column :

⎛ 1⎞ Column I Column II
of the parabola = 4ax is a ⎜ t + ⎟
y2 which
⎝ t⎠ (A) The length of the latus ractum (P) 2
is minimum when t = ± 1. of y2 + 2x + 2by + α = 0, equals
CA

(B) If the tangents from point (0, 2) (Q) – 2

Q.14 Assertion (A) : If 4 and 3 are length of two to y2 = 4ax are inclined at an
focal segment of a focal chord of parabola angle 3π/4, then a =
then latus rectum of this parabola will be (C) A tangent at point A of the (R) –1
48 circle 2(x2 + y2) – 3x + 4y = 0
.
7 pass through the point P (2, 1).
Reason (R) : If l1 and l2 are lengths of Then PA is equal to
(D) One of the lines of (S) 1
segments of a focal chord of a parabola, my2 + (1 – m2) xy – mx2 = 0 is a
2l1l 2 bisector of the angle between the
then its latus rectum is .
l1 + l 2 lines xy = 0, then m =

184 | Parabola ________________________________________________________________________

Q.19 The equation of conics represented by the Passage # 2 (Q.23 to 25)
general equation of second degree y = x is tangent to the parabola y = ax2 + c
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 and the
discriminant of above equation is Q.23 If a = 2, then the value of c is
represented by Δ, where 1 1
(A) (B) –
Δ = abc + 2fgh – af2 – bg2 – ch2 8 2
a h g 1
(C) (D) 1
or Δ= h b f 2
g f c
Q.24 If (1, 1) is point of contact then a is

T
Column I Column II
(A) The conic is represented by (P) degenerate 1 1
(A) (B)
the equation 2 3

IN
⎛x⎞ ⎛ y⎞ 1 1
⎜ ⎟ + ⎜ ⎟ = 1 (a ≠ 0, b ≠ 0) (C) (D)
4 6
⎝a⎠ ⎝b⎠
(B) The conic is represented by (Q) Non-degenerate
the equation 3x2 + 10xy + Q.25 If c = 2, then point of contact is

PO
3y2 – 15x – 21y + 18 = 0 is (A) (2, 2) (B) (4, 4)
(C) The conic is represented by (R) a parabola (C) (6, 6) (D) (3, 3)
the equation 8x2 – 4xy + 5y2
– 16x – 14y + 17 = 0 is (S) an ellipse Part-E : Numeric Response Type Questions
(T) a pair of
intersecting Q.26 An equilateral triangle is inscribed in the
straight lines parabola y2 = 4x with one vertex at the
R
origin. The radius of the circum circle of
Part-D : Passage based objective questions that triangle is
EE

Passage # 1 (Q.20 to 22)

The coordinates of the vertex of the Q.27 M is the foot of the ⊥ from a point P on the
parabola f(x) = 2x2 + px + q are (– 3, 1) parabola y2 = 8(x – 3) to its directrix and S
is the focus of the parabola and SPM is an
Q.20 The value of p is equilateral triangle, then the length of
R

(A) 12 (B) – 12 each side of the triangle is

(C) 19 (D) – 19

Q.28 If the normal at three points (ap2, 2ap),

CA

Q.21 The value of q is

(aq2, 2aq) and (ar2, 2ar) are concurrent then
(A) – 19 (B) 19
(C) – 12 (D) None of these the common root of equations px2 + qx + r = 0
and a(b – c) x2 + b(c – a) x + c(a – b) = 0 is
Q.22 The parabola
(A) touches the x-axis Q.29 Normals of parabola y2 = 4x at P and Q
(B) intersecting the x-axis in two real and meets at R(x2, 0) and tangents at P and Q
distinct points meets at T(x1, 0). If x2 = 3, then find the
(C) lies completely above the x-axis area of quadrilateral PTQR.
(D) lies completely below the x-axis

_________________________________________________________________________ Parabola | 185

Part-F : Subjective Type Questions Q.39 If a2 > 8b2, prove that a point can be found
such that the two tangents from it to the
parabola y2 = 4ax are normals to the
Q.30 Find the latus rectum, the vertex, the
parabola x2 = 4by.
focus, equations of the axis, the directrix
and the tangent at the vertex of the Q.40 The tangents from the point T to the
parabola y2 – 8y – 2x + 9 = 0. parabola y2 = 4ax touch at P and Q. If the
chord of contact PQ is a normal to the
Q.31 Find the number of points with integral parabola at P, prove that TP is bisected by
coordinates (2a, a – 1) that fall in the the directrix of the parabola.
interior of the larger segment of the circle

T
x2 + y2 = 25 cut off by the parabola x2 + 4y = 0. Q.41 If from the vertex of the parabola y2 = 4ax a
Also find their coordinates. pair of chords be drawn at right angles to
one another and with these chords as

IN
Q.32 Prove that in the parabola y2 = 4ax, the adjacent sides a rectangle be drawn. Prove
length of the chord passing through the that the locus of the vertex of the farther
vertex and inclined to the x-axis at an angle angle of the rectangle is the parabola
θ is (4a cos θ)/sin2 θ. y2 = 4a (x – 8a).

PO
Q.33 From the focus of the parabola y2 = 2px as Q.42 A tangent is drawn at any point P on the
centre a circle is described so that a parabola y2 – 2y – 4x + 5 = 0, which meets
common chord of the curves is equidistant the directrix at Q. Find the locus of point R
from the vertex and the focus of the 1
which divides QP externally in : 1.
parabola. Write the equation of the circle. 2

Q.34 Find the equation of common tangents to Q.43 Find the equation of common tangent to
R
the circle x2 + y2 =1 and the parabola the parabolas y2 = 4ax and x2 = 4by.
y2 = 4x, if any such tangent exists.
Q.44 Find the equation of a common tangent to the
Q.35 A variable tangent to the parabola = 4ax
y2 parabola y2 = 2x and the circle x2 + y2 + 4x = 0.
EE

meets the circle x2 + y2 = r2 at P and Q.

Prove that the locus of the mid point of PQ Q.45 Find the condition that the parabolas
is x(x2 + y2) + ay2 = 0. y2 = 4c(x – d) and y2 = 4ax have a common
normal other than x-axis (a > c > 0).
Q.36 Find the equation of the normal to the
R

parabola y2 = 4x at the point (4, 4). Also Q.46 AB, AC are tangents to a parabola y2 = 4ax.
find the point on this normal from which If l 1 , l 2 , l 3 are the length of perpen-
the other two normals drawn to the diculars from A, B, C on any tangent to the
CA

parabola will be at right angles. parabola, then prove that l 2 , l 1 , l 3 are in

GP.
Q.37 A circle cuts a parabola in four points.
Prove that the common chords are in pairs
equally inclined to the axis of the parabola.

Q.38 Find the equation of the parabola whose

axis is along x-axis and which touches the
pair of lines x2 – y2 – 2x + 1= 0, focus being
at (4, 0).

186 | Parabola ________________________________________________________________________

 EXERCISE (Level-4)
Old Examination Questions
Section-A [JEE Main] Q.7 Statement-1 % y = mx −
1
is always a
m
Q.1 The locus of the vertices of the family of tangent to the parabola, y2 = –4x for all
non-zero values of m.
a3x 2 a 2x
parabolas y = + – 2a is Statement-2 % Every tangent to the
3 2 parabola, y2 = –4x will meet its axis at a
[AIEEE 2006] point whose abscissa is non-negative.
[AIEEE Online -2012]

T
3 35
(A) xy = (B) xy = (A) Statement-1 is true, Statement-2 is
4 16
true and Statement-2 is the correct
64 105 explanation of Statement-1

IN
(C) xy = (D) xy =
105 64 (B) Statement-1 is true, Statement-2 is
true and Statement-2 is not the correct
explanation of statement-1
Q.2 The equation of a tangent to the parabola
(C) Statement-1 is true, Statement-2 is
y2 = 8x is y = x + 2. The point on this line false

PO
from which the other tangent to the (D) Statement-1 is false, Statement-2 is
parabola is perpendicular to the given true
tangent is- [AIEEE 2007]
Q.8 The equation of the normal to the parabola,
(A) (–1, 1) (B) (0, 2) (C) (2, 4) (D) (–2, 0) x2 = 8y at x = 4 is :
[AIEEE Online - 2012]
Q.3 A parabola has the origin as its focus and (A) x + 2y = 0 (B) x + y = 2
the line x = 2 as the directrix. Then the (C) x – 2y = 0 (D) x + y = 6
vertex of the parabola is at - [AIEEE 2008] Q.9 The area of the triangle formed by the lines
R
(A) (1, 0) (B) (0, 1) (C) (2, 0) (D) (0, 2) joining the vertex of the parabola, x2 = 8y
to the extremities of its latus rectum is :
Q.4 If two tangents drawn from a point P to the [AIEEE Online - 2012]
EE

(A) 4 (B) 2 (C) 1 (D) 8

parabola y2 = 4x are at right angles, then
the locus of P is - [AIEEE 2010] Q.10 Given : A circle, 2x2 + 2y2 = 5 and a
(A) x = 1 (B) 2x + 1 = 0 parabola, y2 = 4 5 x.
(C) x = –1 (D) 2x – 1 = 0 Statement-I : An equation of a common
tangent to these curves is y = x + 5 .
R

Q.5 The shortest distance between line y – x = 1 5

and curve x = y2 is - [AIEEE 2011] Statement-II : If the line, y = mx +(m ≠ 0)
m
3 3 2 8 4 is their common tangent, then m satisfies
CA

(A) (B) (C) (D)

4 8 3 2 3 m4 – 3m2 + 2 = 0, [JEE Main - 2013]
(A) Statement-I is true; Statement-II is
false.
Q.6 The chord PQ of the parabola y2 = x, where (B) Statement-I is false; Statement-II is
one end P of the chord is at point (4, –2), is true.
perpendicular to the axis of the parabola. (C) Statement-I is true; Statement-II is
true; Statement-II is a correct
Then the slope of the normal at Q is %
explanation for Statement-I.
[AIEEE Online -2012] (D) Statement-I is true; Statement-II is
1 1 true; Statement-II is not a correct
(A) − (B) – 4 (C) (D) 4 explanation for Statement-I.
4 4

_________________________________________________________________________ Parabola | 187

Q.11 Statement-1 : The line x – 2y = 2 meets Q.16 Let L1 be the length of the common chord
the parabola, y2 + 2x = 0 only at the point of the curves x2 + y2 = 9 and y2 = 8x, and L2
(–2, –2). be the length of the latus rectum of y2 = 8x,
then : [JEE Main Online - 2014]
1
Statement-2 : The line y = mx – (m ≠ 0) (A) L1 > L2 (B) L1 = L2
2m
L
is tangent to the parabola, y2 = – 2x at the (C) L1 < L2 (D) 1 = 2
L2
⎛ 1 1⎞
point ⎜ − 2
,− ⎟ .
⎝ 2m m⎠ Q.17 Let O be the vertex and Q be any point on the
[JEE Main Online - 2013] parabola, x2 = 8y. If the point P divides the
(A) Statement-1 is true, Statement-2 is line segment OQ internally in the ratio 1 : 3,

T
then the locus of P is : [JEE Main - 2015]
false
(A) x2 = y (B) y2 = x (C) y2 = 2x (D) x2 = 2y
(B) Statement-1 is true, Statement-2 is

IN
true, Satement-2 is correct explanation Q.18 Let PQ be a double ordinate of the
for Statement-1 parabola, y2 = – 4x, where P lies in the
(C) Statement-1 is false, Statement-2 is second quadrant. If R divides PQ in the
true ratio 2 : 1, then the locus of R is :
(D) Statement-1 is true, Statement-2 is [JEE Main Online - 2015]

PO
true, Statement-2 is not a correct (A) 9y2 = 4x (B) 9y2 = – 4x
(C) 3y2 = 2x (D) 3y2 = – 2x
explanation for Statement-1
Q.19 The centres of those circles which touch the
Q.12 The point of intersection of the normals to circle, x2 + y2 – 8x – 8y – 4 = 0, externally
the parabola y2 = 4x at the ends of its latus and also touch the x-axis, lie on :
rectum is : [JEE Main Online - 2013] [JEE Main - 2016]
(A) (0, 2) (B) (3, 0) (C) (0, 3) (D) (2, 0) (A) a circle
R
(B) an ellipse which is not a circle
(C) a hyperbola
Q.13 The slope of the line touching both the
(D) a parabola
parabolas y2 = 4x and x2 = – 32y is
EE

[JEE Main - 2014]

Q.20 Let P be the point on the parabola, y2 = 8x
2 1 3 1
(A) (B) (C) (D) which is at a minimum distance from the
3 2 2 8 centre C of the circle, x2 + (y + 6)2 = 1. Then
the equation of the circle, passing through
Q.14 A chord is drawn through the focus of the C and having its centre at P is -
R

parabola y2 = 6x such that its distance from [JEE Main - 2016]

5 (A) x2 + y2 – 4x + 8y + 12 = 0
the vertex of this parabola is , then its
2 (B) x2 + y2 – x + 4y – 12 = 0
CA

slope can be ; [JEE Main Online - 2014] x

(C) x2 + y2 – + 2y – 24 = 0
5 3 2 2 4
(A) (B) (C) (D) (D) x2 + y2 – 4x + 9y + 18 = 0
2 2 5 3
Q.21 P and Q are two distinct points on the
Q.15 Two tangents are drawn from a point (–2, –1) parabola, y2 = 4x, with parameters t and t1
to the curve, y2 = 4x. If α is the angle respectively. If the normal at P passes
between them, then |tan α| is equal to :
through Q, then the minimum value of t12 is :
[JEE Main Online - 2014]
1 1 [JEE Main Online- 2016]
(A) (B) (C) 3 (D) 3 (A) 6 (B) 2 (C) 4 (D) 8
3 3

188 | Parabola ________________________________________________________________________

Q.22 If the common tangents to the parabola Q.28 Let P be a point on the parabola, x2 = 4y. If
x2 = 4y and the circle, x2 + y2 = 4 intersect the distance of P from the centre of the
circle x2 + y2 + 6x + 8 = 0 is minimum, then
at the point P, then the distance of P from
the equation of the tangent to the parabola
the origin, is - [JEE Main Online- 2017] at P, is - [JEE-Main Online-2018]
(A) 2( 2 +1) (B) 3 + 2 2 (A) x + 4y – 2 = 0 (B) x + 2y = 0
(C) x + y + 1 = 0 (D) x – y + 3 = 0
(C) 2 +1 (D) 2(3+ 2 2 )
Q.29 Axis of a parabola lies along x-axis. If its
Q.23 If y = mx + c is the normal at a point on the vertex and focus are at distances 2 and 4
respectively from the origin, on the positive
parabola y2 = 8x whose focal distance is
x-axis then which of the following points

T
8 units, then |c| is equal to - does not lie on it ? [JEE Main - 2019]
[JEE Main Online- 2017] (A) (8, 6) (B) (4, –4)
(C) (6, 4 2 ) (D) (5, 2 6 )

IN
(A) 8 3 (B) 10 3 (C) 2 3 (D) 16 3
Q.30 Equation of a common tangent to the circle
Q.24 If the tangent at (1, 7) to the curve = y – 6
x2 x2 + y2 – 6x = 0 and the parabola, y2 = 4x, is :
[JEE Main - 2019]
touches the circle x2 + y2 + 16x + 12y + c = 0
then the value of c is : [JEE Main - 2018] (A) 3y = x + 3 (B) 3y = 3x + 1

PO
(A) 195 (B) 185 (C) 85 (D) 95 (C) 2 3y = –x – 12 (D) 2 3y = 12x + 1

Q.31 Let A(4, –4) and B(9, 6) be points on the

Q.25 Tangent and normal are drawn at P(16, 16) parabola, y2 = 4x. Let C be chosen on the
on the parabola y2 = 16x, which intersect arc AOB of the parabola, where O is the
origin, such that the area of ΔACB is
the axis of the parabola at A and B,
maximum. Then, the area (in sq. units) of
respectively. If C is the centre of the circle ΔACB, is : [JEE Main - 2019]
R
through the points P, A and B and ∠CPB = θ, 3 1 1
(A) 32 (B) 31 (C) 30 (D) 31
then a value of tan θ is : [JEE Main - 2018] 4 2 4
1 4
(A) (B) 2 (C) 3 (D) Q.32 If the parabolas y2 = 4b(x – c) and y2 = 8ax
EE

2 3 have a common normal, then which on of

the following is a valid choice for the
Q.26 Two parabolas with a common vertex and ordered triad (a, b, c) ?
with axes along x-axis and y-axis, [JEE Main - 2019]
(A) (1, 1, 3) (B) (1, 1, 0)
respectively, intersect each other in the
⎛1 ⎞ ⎛1 ⎞
R

first quadrant. If the length of the latus (C) ⎜ , 2, 0 ⎟ (D) ⎜ , 2, 3 ⎟

rectum of each parabola is 3, then the ⎝2 ⎠ ⎝2 ⎠
equation of the common tangent to the two Q.33 The length of the chord of the parabola
CA

parabolas is [JEE-Main Online-2018] x2 = 4y having equation x – 2y + 4 2 = 0 is –

(A) 3(x + y) + 4 = 0 (B) 8(2x + y) + 3 = 0 [JEE Main - 2019]
(C) 4(x + y) + 3 = 0 (D) x + 2y + 3 = 0
(A) 8 2 (B) 6 3 (C) 3 2 (D) 2 11

Q.27 Tangents drawn from the point (–8, 0) to Q.34 If the area of the triangle whose one vertex
the parabola y2 = 8x touch the parabola at is at the vertex of the parabola,
P and Q. If F is the focus of the parabola, y2 + 4(x – a2) = 0 and the other two vertices
then the area of the triangle PFQ (in sq. are the points of intersection of the
parabola and y-axis, is 250 sq. units, then a
units) is equal to -
value of 'a' is [JEE Main - 2019]
[JEE-Main Online-2018]
(A) 5 5 (B) (10)2/3 (C) 5(21/3) (D) 5
(A) 48 (B) 32 (C) 24 (D) 64
_________________________________________________________________________ Parabola | 189
Q.35 The maximum area (in sq. units) of a Q.42 The locus of the mid-point of the line
rectangle having its base on the x-axis and segment joining the focus of the parabola
its other two vertices on the parabola, y2 = 4ax to a moving point of the parabola,
y = 12 – x2 such that the rectangle lies is another parabola whose directrix is :
inside the parabola, is : [JEE Main - 2019] [JEE Main - 2021]
a a
(A) 36 (B) 20 2 (C) 18 3 (D) 32 (A) x =– (B) x = (C) x = 0 (D) x = a
2 2

Q.36 Let P(4, –4) and Q(9, 6) be two points on the Q.43 Let C be the locus of the mirror image of a
parabola, y2 = 4x and let x be any point on point on the parabola y2 = 4x with respect to
the arc POQ of this parabola, where O is the
the line y = x. Then the equation of tangent
vertex of this parabola, such that the area of

T
to C at P(2, 1) is : [JEE Main - 2021]
ΔPXQ is maximum. Then this maximum
area (in sq. units) is : [JEE Main - 2019] (A) x – y = 1 (B) 2x + y = 5
625 125 75 125 (C) x + 3y = 5 (D) x + 2y = 4

IN
(A) (B) (C) (D)
4 4 2 2
Section-B [JEE Advanced]
Q.37 The equation of a tangent to the parabola,
x2 = 8y, which makes an angle θ with the Q.1 A tangent at any point P (1, 7) the parabola
positive directions of x-axis, is : y = x2 + 6, which is touching to the circle

PO
[JEE Main - 2019] x2 + y2 + 16x + 12y + c = 0 at point Q, then Q
(A) x = y cot θ – 2 tan θ is [IIT-Screening -2005]
(B) y = x tan θ + 2 cot θ (A) (–6, –7) (B) (–10, –15)
(C) x = y cot θ + 2 tan θ (C) (–9, –7) (D) (–6, –3)
(D) y = x tan θ – 2 cot θ
Q.38 The tangent to the parabola y2 = 4x at the Q.2 Locus of the centre of circle touching to the
point where it intersects the circle x2 + y2 = 5 x-axis & the circle x2 + (y − 1)2 = 1
in the first quadrant, passes through the externally is- [IIT SCR.-2005]
R
point - [JEE Main - 2019] (A) {(0, y) ; y ≤ 0} ∪ (x = 4y)
2

⎛ 1 4⎞ ⎛3 7⎞ (B) {(0, y) ; y ≤ 0} ∪ (x2 = y)

(A) ⎜ − , ⎟ (B) ⎜ , ⎟
⎝ 3 3⎠ ⎝4 4⎠ (C) {(x, y) ; 0 ≤ y} ∪ (x2 = 4y)
EE

⎛1 3⎞ ⎛ 1 1⎞ (D) {(0, y) ; y ≥ 0} ∪ {x2 + (y − 1)2 = 4}

(C) ⎜ , ⎟ (D) ⎜ − , ⎟
⎝4 4⎠ ⎝ 4 2⎠
Q.3 The axis of parabola is along the line y = x
Q.39 If one end of a focal chord of the parabola, and the distance of vertex from origin is √2
y2 = 16x is at (1, 4), then the length of this and that from its focus is 2√2. If vertex and
focal chord is : [JEE Main - 2019]
focus both lie in the first quadrant, so the
R

(A) 24 (B) 20 (C) 25 (D) 22

equation of parabola is [IIT- 2006]
Q.40 If y = mx + 4 is a tangent to both the (A) (x – y) 2 = 16(x + y –2)
parabolas, y2 = 4x and x2 = 2by, then b is (B) (x – y) 2 = 4(x + y –2)
CA

equal to [JEE Main - 2020] (C) (x – y) 2 = (x + y –2)

(A) –64 (B) –128 (C) –32 (D) 128 (D) (x – y) 2 = (x – y –2)

Q.41 If one end of a focal chord AB of the Q.4 The equation(s) of common tangent(s) to
⎛1 ⎞ the parabola y = x2 and y = – (x – 2)2
parabola y2 = 8x is at A ⎜ , − 2 ⎟ , then the
⎝ 2 ⎠ MC [IIT-2006]
equation of the tangent to it at B is – (A) y = –4 (x –1)
[JEE Main - 2020] (B) y = 0
(A) 2x + y – 24 = 0 (B) x + 2y + 8 = 0 (C) y = 4 (x – 1)
(C) x – 2y + 8 = 0 (D) 2x – y – 24 = 0 (D) y = –30x – 50

190 | Parabola ________________________________________________________________________

Q.5 Three normals drawn at P,Q and R on the Q.9 The ratio of the areas of the triangles PQS
parabola y2 = 4x intersect at (3, 0). Then and PQR is -
[IIT-2006] (A) 1 : 2 (B) 1 : 2
Column 1 Column 2 (C) 1 : 4 (D) 1 : 8
5
(A) Radius of circumcircle (P)
2 Q.10 The radius of the circumcircle of the
of ΔPQR triangle PRS is -
⎛5 ⎞ (A) 5 (B) 3 3 (C) 3 2 (D) 2 3
(B) Area of ΔPQR (Q) ⎜ , 0 ⎟
⎝2 ⎠
⎛2 ⎞ Q.11 The radius of the incircle of the triangle
(C) Centroid of ΔPQR (R) ⎜ , 0 ⎟ PQR is
⎝3 ⎠

T
8
(D) Circumcentre of ΔPQR (S) 2 (A) 4 (B) 3 (C) (D) 2
3

IN
Passage (Q. 6 to 8) Q.12 Consider the two curves [IIT-2008]
Let C1 is a circle touching to all the sides of C1 : y2 = 4x
square ABCD of side length 2 units C2 : x2 + y2 – 6x + 1 = 0
internally and C2 circle is passing through Then,
the vertices of square. A line L is drawn (A) C1 and C2 touch each other only at one
through A. [IIT 2006] point

PO
(B) C1 and C2 touch each other exactly at
Q.6 Let P is a point on C1 and Q is on C2, then two points
PA 2 + PB 2 + PC 2 + PD2 (C) C1 and C2 intersect (but do not touch) at
=
QA 2 + QB 2 + QC 2 + QD2 exactly two points
(A) 0.75 (B) 0.5 (C) 1.25 (D) 1 (D) C1 and C2 neither intersect nor touch
each other
Q.7 A variable circle touches to the line L and
circle C1 externally such that both circles Q.13 The tangent PT and the normal PN to the
R
are on the same side of the line, then the parabola y2 = 4ax at a point P on it meet its
locus of center of variable circle is axis at points T and N, respectively. The
(A) Ellipse (B) Circle locus of the centroid of the triangle PTN is
EE

(C) Hyperbola (D) Parabola

a parabola whose : [IIT-2009]
Q.8 A line M through A is drawn parallel to ⎛ 2a ⎞
(A) vertex is ⎜ ,0 ⎟
BD. Locus of point R, which moves such ⎝ 3 ⎠
that its distances from the line BD and the (B) directrix is x = 0
vertex A are equal, cuts to line M at
2a
R

T2 and T3 and AC at T1, then area of (C) latus rectum is

triangle T1 T2 T3 is- 3
1 (D) focus is (a, 0)
(A) (sq.units) (B) 2 (sq.units)
CA

2
4 Q.14 Let A and B be two distinct points on the
(C) 1 (sq.units) (D) (sq.units)
3 parabola y2 = 4x. If the axis of the parabola
touches a circle of radius r having AB as its
Passage (Q. 9 to 11)
Consider the circle x2 + y2 = 9 and the diameter, then the slope of the line joining
parabola y2 = 8x. They intersect at P and Q A and B can be - M [IIT-2010]
in the first and the fourth quadrants, 1 1
respectively. Tangents to the circle at P (A) − (B)
r r
and Q intersect the x-axis at R and
2 2
tangents to the parabola at P and Q (C) (D) −
intersect the x-axis at S. [IIT 2007] r r

_________________________________________________________________________ Parabola | 191

Q. 15 Let (x, y) be any point on the parabola Q.21 A line L : y = mx + 3 meets y-axis at
y2 = 4x. Let P be the point that divides the E (0, 3) and the arc of the parabola
line segment from (0, 0) to (x, y) in the ratio y2 = 16x, 0 ≤ y ≤ 6 at the point F(x0, y0). The
1 : 3. Then the locus of P is [IIT-2011] tangent to the parabola at F(x0, y0)
(A) x = y
2 (B) y = 2x
2 intersects the y-axis at G(0, y1). The slope
(C) y = x
2 (D) x2 = 2y m of the line L is chosen such that the area
of the triangle EFG has a local maximum.
Q.16 Let L be a normal to the parabola y2 = 4x. Match List-I with List-II and select the
If L passes through the point (9, 6), then L correct answer using the code given below
is given by MCQ [IIT-2011] the lists : [JEE-Advanced 2013]

T
(A) y – x + 3 = 0 (B) y + 3x – 33 = 0 List-I List-II
(C) y + x – 15 = 0 (D) y – 2x + 12 = 0 1
(P) m = (1)

IN
2
Q.17 Consider the parabola y2 = 8x. Let Δ1 be the (Q) Maximum area of ΔEFG is (2) 4
area of the triangle formed by the end (R) y0 = (3) 2
points of its latus rectum and the point (S) y1 = (4) 1
⎛1 ⎞ Codes :
P ⎜ , 2 ⎟ on the parabola, and Δ2 be the area

PO
⎝2 ⎠ P Q R S
of the triangle formed by drawing tangents (A) 4 1 2 3
at P and at the end points of the latus (B) 3 4 1 2
Δ1 (C) 1 3 2 4
rectum. Then is [IIT-2011]
Δ2 (D) 1 3 4 2

Q.22 The common tangents to the circle x2 + y2 = 2

Q.18 Let S be the focus of the parabola y2 = 8x
R
and the parabola y2 = 8x touch the circle at
and let PQ be the common chord of the
circle x2 + y2 – 2x – 4y = 0 and the given the points P, Q and the parabola at the
parabola. The area of the triangle PQS is points R, S. Then the area of the
EE

[IIT-2012] quadrilateral PQRS is

[JEE-Advanced 2014]
Paragraph for Questions 19 and 20 (A) 3 (B) 6 (C) 9 (D) 15
Let PQ be a focal chord of the parabola
y2 = 4ax. The tangents to the parabola at P Paragraph For Questions 23 and 24
R

and Q meet at a point lying on the line Let a, r, s, t be nonzero real numbers. Let
y = 2x + a, a > 0. [JEE-Advanced 2013] P(at2, 2at), Q, R(ar2, 2ar) and S(as2, 2as) be
distinct points on the parabola y2 = 4ax.
CA

Q.19 If chord PQ subtends an angle θ at the Suppose that PQ is the focal chord and
vertex of y2 = 4ax, then tan θ = lines QR and PK are parallel, where K is
2 −2 the point (2a, 0).
(A) 7 (B) 7
3 3
2 −2 Q.23 The value of r is [JEE-Advanced 2014]
(C) 5 (D) 5
3 3 1 t2 + 1
(A) – (B)
t t
Q.20 Length of chord PQ is
1 t2 – 1
(A) 7a (B) 5a (C) 2a (D) 3a (C) (D)
t t

192 | Parabola ________________________________________________________________________

Q.24 If st = 1, then the tangent at P and the (A) Q2Q3 = 12
normal at S to the parabola meet at a point (B) R2R3 = 4 6
whose ordinate is [JEE-Advanced 2014] (C) Area of the triangle OR2R3 is 6 2
2 2 2 2
(t + 1) a(t + 1) (D) Area of the triangle PQ2Q3 is 4 2
(A) 3
(B)
2t 2t 3
a(t 2 + 1)2 a(t 2 + 2) 2 Q.29 Let P be the point on the parabola y2 = 4x
(C) (D)
t3 t3 which is at the shortest distance from the
center S of the circle x2 + y2 – 4x – 16y + 64 = 0.
Q.25 If the normals of the parabola y2 = 4x drawn Let Q be the point on the circle dividing the
line segment SP internally. Then

T
at the end points of its latus rectum are
tangents to the circle (x – 3)2 + (y + 2)2 = r2, MCQ [JEE - Advanced 2016]

then the value of r2 is (A) SP = 2 5

IN
[JEE - Advanced 2015] (B) SQ : QP = ( 5 + 1) : 2
(C) the x-intercept of the normal to the
Q.26 Let the curve C be the mirror image of the parabola at P is 6
parabola y2 = 4x with respect of the line (D) the slope of the tangent to the circle at

PO
x + y + 4 = 0. If A and B are the points of 1
Q is
intersection of C with the line y = –5, then 2
the distance between A and B is
[JEE - Advanced 2015] Q.30 If a chord, which is not a tangent, of the
parabola y2 = 16x has the equation 2x + y = p,
Q.27 Let P and Q be distinct points on the and midpoint (h, k), then which of the
following is(are) possible value(s) of p, h and
parabola y2 = 2x such that a circle with PQ as
k? [JEE - Advanced 2017]
R
diameter passes through the vertex O of the (A) p = 5, h = 4, k = – 3
parabola. If P lies in the first quadrant and (B) p = 2, h = 3, k = – 4
the area of the triangle ΔOPQ is 3 2 , then (C) p = –2, h = 2, k = – 4
EE

(D) p = –1, h = 1, k = – 3
which of the following is (are) the coordinates
of P ? [JEE - Advanced 2015] Q.31 Let the circles C1 : x2 + y2 = 9 and
(A) (4, 2 2 ) (B) (9, 3 2 ) C2 : (x – 3)2 + (y – 4)2 = 16, intersect at the
points X and Y. Suppose that another circle
R

⎛1 1 ⎞
(C) ⎜⎜ , ⎟⎟ (D) (1, 2 ) C3 : (x – h)2 + (y – k)2 = r2 satisfies the
⎝4 2 ⎠ following conditions.
(i) centre of C3 is collinear with the
CA

Q.28 The circle C1 : x2 + y2 = 3, with centre at O, centres of C1 and C2,

intersects the parabola x2 = 2y at the point (ii) C1 and C2 both lie inside C3, and
P in the first quadrant. Let the tangent to (iii) C3 touches C1 at M and C2 at N.
Let the line through X and Y intersect C3
the circle C1 at P touches other two circles
at Z and W, and let a common tangent of
C2 and C3 at R2 and R3, respectively.
C1 and C3 be a tangent to the parabola
Suppose C2 and C3 have equal radii 2 3 x2 = 8αy.
and centre Q2 and Q3, respectively. If Q2 There are some expressions given in the
List-I whose values are given in List-II
and Q3 lie on the y-axis, then
below : [JEE - Advanced 2019]
M [JEE - Advanced 2016]

_________________________________________________________________________ Parabola | 193

 LIST-I LIST-II Which of the following is the only
(I) 2h + k (P) 6 CORRECT combination?
(A) (II), (Q) (B) (II), (T)
(II) Length of ZW (Q) 6 (C) (I), (S) (D) (I), (U)
Length of XY
(III) Area of triangle MZN (R) 5 Q.33 Let a, b and λ be positive real numbers.
Area of triangle ZMW 4 Suppose P is an end point of the latus
(IV) α (S) 21 rectum of the parabola y2 = 4λx, and
5 x2 y2
suppose the ellipse + = 1 pass
(T) a2 b2
2 6
through the point P. If the tangents to the

T
(U) 10 parabola and the ellipse at the point P are
3 perpendicular to each other, then the

IN
Which of the following is the only eccentricity of the ellipse is-
INCORRECT combination? [JEE - Advanced 2019]
1 1 1 2
(A) (IV), (U) (B) (I), (P) (A) (B) (C) (D)
2 2 3 5
(C) (IV), (S) (D) (III), (R)

PO
Q.34 Let E denote the parabola y2 = 8x. Let P =
Q.32 Let the circles C1 : x2 + y2 = 9 and
(–2, 4) and let Q and Q′ be two distinct
C2 : (x – 3)2 + (y – 4)2 = 16, intersect at the
points on E such that the lines PQ and PQ′
points X and Y. Suppose that another circle are tangents to E. Let F be the focus of E.
C3 : (x – h)2 + (y – k)2 = r2 satisfies the Then which of the following statements is
(are) TRUE? [JEE - Advanced 2021]
following conditions
(A) The triangle PFQ is a right-angled
(i) centre of C3 is collinear with the
R
centres of C1 and C2, triangle
(ii) C1 and C2 both lie inside C3, and (B) The triangle QPQ′ is a right-angled
(iii) C3 touches C1 at M and C2 at N. triangle
EE

Let the line through X and Y intersect C3

at Z and W, and let a common tangent of (C) The distance between P and F is 5 2
C1 and C3 be a tangent to the parabola (D) F lies on the line joining Q and Q′
x2 = 8αy. There are some expressions given
in the List-I whose values are given in List-II
below : [JEE - Advanced 2019]
R

LIST-I LIST-II
(I) 2h + k (P) 6
(II) Length of ZW (Q) 6
CA

Length of XY
(III) Area of triangle MZN (R) 5
Area of triangle ZMW 4
(IV) α (S) 21
5
(T) 2 6
(U) 10
3

194 | Parabola ________________________________________________________________________

 EXERCISE (Level-5)
Review Exercise
Q.1 If (a2, a – 2) be a point interior to the Q.7 Find the locus of the point of intersection of
region of the parabola y2 = 2x bounded by those normals to the parabola x2 = 8y
the chord joining the points (2, 2) & (8, –4) which are at right angles to each other.
then find the set of all possible real values
[IIT 1997]
of a.

Q.2 Three normals are drawn from the point Q.8 Let C1 and C2 be, respectively, the
(c, 0) to the curve y2 = x. Show that c must parabolas x2 = y – 1 and y2 = x – 1. Let P be

T
be greater than 1/2. One normal is always any point on C1 and Q be any point on C2.
the x-axis. Find c for which the other two Let P1 and Q1 be the reflections of P and Q,
normals are perpendicular to each other.

IN
respectively, with respect to the line y = x.
[IIT 1991]
Prove that P1 lies on C2, Q1 lies on C1 and
Q.3 Through the vertex O of parabola y2 = 4x, PQ > min {PP1, QQ1}. Hence or otherwise
chords OP and OQ are drawn at right determine points P0 and Q0 on the parabola
angles to one another. Show that all C1 and C2 respectively such that P0Q0 ≤ PQ

PO
positions of P, PQ cuts the axis of the for all pairs (P, Q) with P on C1 and Q on
parabola at a fixed point. Also find the C2. [IIT 2000]
locus of the middle point of PQ.
[IIT 1994]
Q.9 Normals with slopes m1, m2 & m3 are
Q.4 Show that the locus of a point that divides drawn from a point P, not on the axes, to
a chord of slope 2 of the parabola y2 = 4x
the parabola y2 = 4x. If the locus of P under
internally in the ratio 1 : 2 is a parabola.
Find the vertex of this parabola. the condition m1m2 = α is a part of the
parabola, determine the value of α.
R
[IIT 1995]
[IIT 2003]
Q.5 A ray of light is coming along the line y = b
from the positive direction of x-axis & Q.10 Two parabolas C and D intersect at the two
EE

strikes a concave mirror whose intersection different points, where C is y = x2 – 3 and D

with the xy plane is a parabola y2 = 4ax.
is y = kx2. The intersection at which the
Find the equation of the reflected ray &
show that it passes through the focus of the x-value is positive is designated point A,
parabola. Both a & b are positive. and x = a at this intersection. The tangent
[IIT 1995] line l at A to the curve D intersects curve C
R

at point B, other than A. If x-value of point

Q.6 (a) Points A, B and C lie on the parabola B is 1, then find value of 'a'.
y2 = 4ax. The tangents to the parabola at
A, B & C taken in pairs, intersect at
CA

points P, Q & R. Determine the Q.11 The normal at a point P to the parabola
ratio of the areas of the triangles y2 = 4ax meets its axis at G. Q is another
ABC & PQR. point on the parabola such that QG is
(b) From a point A common tangents perpendicular to the axis of the parabola.
drawn to the circle x2 + y2 = (a2/2) and Prove that QG2 – PG2 = constant.
parabola y2 = 4ax. Find the area of the
quadrilateral formed by the common
tangents, the chord of contact of the
circle & the chord of contact of the
parabola. [IIT 1996]

_________________________________________________________________________ Parabola | 195

 ANSWER KEY
EXERCISE (Level-1)
1. (A) 2. (D) 3. (C) 4. (A) 5. (C) 6. (C) 7. (A)
8. (A) 9. (B) 10. (C) 11. (D) 12. (B) 13. (D) 14. (B)
15. (B) 16. (B) 17. (C) 18. (A) 19. (A) 20. (A) 21. (B)
22. (D) 23. (A) 24. (C) 25. (C) 26. (B) 27. (D) 28. (C)
29. (A) 30. (B) 31. (A) 32. (D) 33. (D) 34. (B) 35. (A)

T
EXERCISE (Level-2)
1. (A) 2. (A) 3. (C) 4. (B) 5. (C) 6. (A) 7. (A)

IN
8. (C) 9. (A) 10. (C) 11. (A) 12. (A) 13. (C) 14. (C)
15. (D) 16. (C) 17. (D) 18. (C) 19. (A) 20. (D) 21. (D)
22. (C) 23. (D) 24. (B) 25. (C) 26. (D) 27. (C) 28. (B)
29. (B) 30. (C) 31. (D) 32. (C) 33. (C) 34. (D) 35. (B)

PO
36. (B) 37. (C) 38. (B) 39. (B) 40. (C)

EXERCISE (Level-3)
Part-A
1. (A,B,C) 2. (A,C) 3. (C) 4. (A,B) 5. (A,B) 6. (A,D) 7. (A)
8. (A,B) 9. (A,B,C) 10. (A,B,C,D) 11. (A,B,C,D) 12. (A,B,D)
R
Part-B
13. (A) 14. (C) 15. (B) 16. (D)
EE

Part-C
17. A → R; B → Q; C → P; D → S 18. A → P; B → P,Q; C → P; D → R, S
19. A → Q,R; B → P,T; C → Q, S

Part-D
R

20. (A) 21. (B) 22. (C) 23. (A) 24. (A) 25. (B)

Part-E
CA

26. 8 27. 8 28. 1 29. 8

Part-F
⎛ 7 ⎞ 7
30. 2, ⎜ − , 4 ⎟ , (–3, 4), y = 4, x = –4 and x = – 31. Two; (2, 0); (4, 1)
⎝ 2 ⎠ 2
2
⎛ p⎞ 9 p2 1 1
33. ⎜ x − ⎟ + y2 = 34. x ± (1 + 5 ) y + (1 + 5 ) = 0 36. y + 2x = 12 ; (7, – 2)
⎝ 2⎠ 16 2 2
38. y2 = 6x – 15 42. 4 + (x + 1) (1 – y)2 = 0 43. a1/3x + b1/3y + (ab)2/3 = 0
44. x ± 2 6 y + 12 = 0 45. 2a < 2c + d

196 | Parabola ________________________________________________________________________

 EXERCISE (Level-4)
SECTION-A
1. (D) 2. (D) 3. (A) 4. (C) 5. (B) 6. (B) 7. (B)
8. (D) 9. (D) 10. (D) 11. (B) 12. (B) 13. (B) 14. (A)
15. (D) 16. (C) 17. (D) 18. (B) 19. (D) 20. (A) 21. (D)
22. (A) 23. (B) 24. (D) 25. (B) 26. (C) 27. (A) 28. (C)
29. (A) 30. (A) 31. (D) 32. (A or All) 33. (B) 34. (D) 35. (D)

T
36. (B) 37. (C) 38. (B) 39. (C) 40. (B) 41. (C) 42. (C)
43. (A)

IN
SECTION-B
1. (A) 2. (A) 3. (A) 4. (B,C) 5. A → P; B → S; C → R; D → Q
6. (A) 7. (D) 8. (C) 9. (C) 10. (B) 11. (D) 12. (B)

PO
13. (A, D) 14. (C, D) 15. (C) 16. (A, B, D) 17. 2 18. 4 19. (D)
20. (B) 21. (A) 22. (D) 23. (D) 24. (B) 25. 2 26. 4
27. (A,D) 28. (A,B,C) 29. (A,C,D) 30. (B) 31. (C) 32. (A) 33. (A)
34. (A,B,D)

EXERCISE (Level-5)
R
3
1. (–2 + 2 2 , 2) 2. c = 3. y2 = 2 (x – 4)
4
2
4 ⎛ 2⎞ ⎛ 8⎞ ⎛2 8⎞
EE

4. ⎜ x − ⎟ = ⎜ y − ⎟ , vertex ⎜ , ⎟ 5. 4abx + (4a2 – b2) y – 4a2 b = 0

9 ⎝ 9⎠ ⎝ 9⎠ ⎝9 9⎠

15a 2
6. (a) 2 : 1 (b) 7. x2 + 2 (4 – y) + 4 = 0
4

⎛1 5 ⎞ ⎛5 1⎞
R

8. P0 = ⎜ , ⎟ , Q0 = ⎜ , ⎟ 9. α = 2 10. 3
⎝2 4⎠ ⎝4 2⎠
CA

_________________________________________________________________________ Parabola | 197

 NOTES

T
IN
PO
R
EE
R
CA

198 | Parabola ________________________________________________________________________