Parabola JEE Main 2024 - 2019 PYQs

Chapter Priority: Priority D MathonGo

Q1. Let A, B and C be three points on the parabola y 2

= 6x and let the line segment AB meet the line L through C
parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and B
09 Apr 2024 (E)
2
on L. Then ( AM ⋅BN

CD
) is equal to _________

Q2^. Consider the circle C : x

2
+ y
2
= 4 and the parabola P : y
2
= 8x . If the set of all values of α, for which three
chords of the circle C on three distinct lines passing through the point (α, 0) are bisected by the parabola P is
the interval (p, q), then (2q − p) is equal to ________ 2
09 Apr 2024 (E)

Q3. Let a conic C pass through the point (4, −2) and P (x, y), x ≥ 3, be any point on C . Let the slope of the line
touching the conic C only at a single point P be half the slope of the line joining the points P and (3, −5). If
the focal distance of the point (7, 1) on C is d, then 12d equals ______ 06 Apr 2024 (M)

Q4*. Let L 1
, L2 be the lines passing through the point P (0, 1) and touching the parabola 9x 2
+ 12x + 18y − 14 = 0

. Let Q and R be the points on the lines L and L such that the △P QR is an isosceles triangle with base QR.
1 2

If the slopes of the lines QR are m and m , then 16 (m 1 2

2
1
+ m )
2
2
is equal to _______ 06 Apr 2024 (M)

Q5. Suppose AB is a focal chord of the parabola y 2

= 12x of length l and slope m < √3. If the distance of the
chord AB from the origin is d, then l d is equal to _______ 2
05 Apr 2024 (M)

Q6. Let a line perpendicular to the line 2x − y = 10 touch the parabola y 2

= 4(x − 9) at the point P . The distance
of the point P from the centre of the circle x 2
+ y
2
− 14x − 8y + 56 = 0 is __________ 05 Apr 2024 (E)

Q7. Let the length of the focal chord PQ of the parabola y 2

= 12x be 15 units. If the distance of PQ from the origin
is p, then 10p is equal to _______
2
04 Apr 2024 (M)

Q8^. Let PQ be a chord of the parabola y 2

= 12x and the midpoint of PQ be at (4, 1). Then, which of the following
point lies on the line passing through the points P and Q? 04 Apr 2024 (E)
(1) (3, −3) (2) (2, −9)
(3) ( 3

2
, −16) (4) ( 1

2
, −20)

Q9. One of the points of intersection of the curves y = 1 + 3x − 2x and y = 2 1

x
is ( 1

2
, 2) . Let the area of the region
enclosed by these curves be 1

24
(l√ 5 + m) − n log e (1 + √ 5) , where l, m, n ∈ N . Then l + m + n is equal to
04 Apr 2024 (M)
(1) 29 (2) 31
(3) 30 (4) 32

Q10. The maximum area of a triangle whose one vertex is at (0, 0) and the other two vertices lie on the curve
y = −2x + 54 at points (x, y) and (−x, y) where y > 0 is : 30 Jan 2024 (M)
2

(1) 88 (2) 122

(3) 92 (4) 108

Q11^. Let P (α, β) be a point on the parabola y 2

= 4x . If P also lies on the chord of the parabola x 2
= 8y whose
mid point is (1, 5

4
, then (α − 28)(β − 8) is equal to _______.
) 29 Jan 2024 (E)

Q12. If the shortest distance of the parabola y 2

= 4x from the centre of the circle x 2
+ y
2
− 4x − 16y + 64 = 0 is d
, then d is equal to :
2
27 Jan 2024 (M)

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Parabola JEE Main 2024 - 2019 PYQs
Chapter Priority: Priority D MathonGo

(1) 16 (2) 24
(3) 20 (4) 36

Q13. Let PQ be a focal chord of the parabola y

2
= 36x of length 100, making an acute angle with the positive x−

axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that
P M : M Q = 3 : 1. Then which of the following points does NOT lie on the line passing through M and
perpendicular to the line P Q?
13 Apr 2023 (M)
(1) (−6, 45) (2) (6, 29)
(3) (3, 33) (4) (−3, 43)

Q14*. Let a common tangent to the curves y 2

= 4x and (x − 4) 2
+ y
2
= 16 touch the curves at the points P and Q.
Then (P Q) is equal to ________.
2
10 Apr 2023 (M)

Q15. Let R be the focus of the parabola y 2

= 20x and the line y = mx + c intersect the parabola at two points P
and Q. Let the points G(10, 10) be the centroid of the triangle P QR. If c − m = 6, then P Q is 2

08 Apr 2023 (M)

(1) 296 (2) 325
(3) 317 (4) 346

Q16. Let A(0, 1), B(1, 1) and C(1, 0) be the mid-points of the sides of a triangle with incentre at the point D. If the
focus of the parabola y 2
= 4ax passing through D is (α + β√2, 0), where α and β are rational numbers, then

β
α
2
is equal to 08 Apr 2023 (E)
(1) 8 (2) 12
(3) 6 (4) 9

Q17*. The ordinates of the points P and Q on the parabola with focus (3, 0) and directrix x = −3 are in the ratio
2

3 : 1 . If R(α, β) is the point of intersection of the tangents to the parabola at P and Q, then β

α
is equal to
08 Apr 2023 (E)

Q18. If the x-intercept of a focal chord of the parabola y 2

= 8x + 4y + 4 is 3, then the length of this chord is equal
to _____ . 01 Feb 2023 (E)

Q19*. Let S be the set of all a ∈ N such that the area of the triangle formed by the tangent at the point
P (b, c), b, c ∈ N , on the parabola y 2
= 2ax and the lines x = b, y = 0 is 16 unit , then ∑ 2
a∈S
a is equal to
_____ . 31 Jan 2023 (E)

Q20. The parabolas: ax 2

+ 2bx + cy = 0 and d 2
+ 2ex + f y = 0 intersect on the line y = 1. If a, b, c, d, e, f are
positive real numbers and a, b, c are in G. P., then 30 Jan 2023 (E)
(1) d, e, f are in A.P. (2) d

a
,
e

b
,
f

c
are in G.P.
(3) d

a
,
e

b
,
f

c
are in A.P. (4) d, e, f are in G.P.

Q21*. Let A be a point on the x-axis. Common tangents are drawn from A to the curves x 2
+ y
2
= 8 and y 2
= 16x .
If one of these tangents touches the two curves at Q and R, then (QR) is equal to 2
30 Jan 2023 (E)

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Parabola JEE Main 2024 - 2019 PYQs
Chapter Priority: Priority D MathonGo

(1) 64 (2) 76
(3) 81 (4) 72

Q22*. If P (h, k) be point on the parabola x = 4y , which is nearest to the point Q(0, 33), then the distance of P
2

from the directrix of the parabola y 2

= 4(x + y) is equal to: 30 Jan 2023 (M)
(1) 2 (2) 4
(3) 8 (4) 6

Q23*. A triangle is formed by the tangents at the point (2, 2) on the curves y 2
= 2x and x 2
+ y
2
= 4x, and the line
x + y + 2 = 0 . If r is the radius of its circumcircle, then r is equal to 2
29 Jan 2023 (E)

Q24*. The equations of two sides of a variable triangle are x = 0 and y = 3, and its third side is a tangent to the
parabola y 2
= 6x . The locus of its circumcentre is : 25 Jan 2023 (E)
(1) 4y 2
− 18y − 3x − 18 = 0 (2) 4y 2
+ 18y + 3x + 18 = 0

(3) 4y 2
− 18y + 3x + 18 = 0 (4) 4y 2
− 18y − 3x + 18 = 0

Q25*. The distance of the point (6, −2√2) from the common tangent y = mx + c, m > 0, of the curves x = 2y 2

and x = 1 + y is 2
25 Jan 2023 (M)
(1) 1

3
(2) 5
(3) 14

3
(4) 5√3

Q26*. The equations of sides AB and AC of a triangle ABC are (λ + 1)x + λy = 4 and λx + (1 − λ)y + λ = 0
respectively. Its vertex A is on the y−axis and its orthocentre is (1, 2). The length of the tangent from the
point C to the part of the parabola y 2
= 6x in the first quadrant is 24 Jan 2023 (E)
(1) √6 (2) 2√2
(3) 2 (4) 4

Q27^. Let a tangent to the curve y

2
= 24x meet the curve xy = 2 at the points A and B. Then the mid-points of
such line segments AB lie on a parabola with the 24 Jan 2023 (M)
(1) directrix 4x = 3 (2) directrix 4x = −3
(3) Length of latus rectum 3

2
(4) Length of latus rectum 2

Q28*. Two tangent lines l and l are drawn from the point (2, 0) to the parabola 2y
1 2
2
= −x . If the lines l and l are
1 2

also tangent to the circle (x − 5) 2

+ y
2
= r , then 17r is equal to
2
28 Jul 2022 (E)

Q29^. If the tangents drawn at the points P and Q on the parabola y 2

= 2x − 3 intersect at the point R(0, 1), then
the orthocentre of the triangle P QR is 28 Jul 2022 (M)
(1) (0, 1) (2) (2, −1)
(3) (6, 3) (4) (2, 1)

Q30*. Let P (a, b) be a point on the parabola y 2

= 8x such that the tangent at P passes through the centre of the
circle x 2
+ y
2
− 10x − 14y + 65 = 0 . Let A be the product of all possible values of a and B be the product
of all possible values of b. Then the value of A + B is equal to 27 Jul 2022 (M)
(1) 0 (2) 25
(3) 40 (4) 65

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Parabola JEE Main 2024 - 2019 PYQs
Chapter Priority: Priority D MathonGo

Q31. If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is x + y = a,
is 16, then |a| is equal to 27 Jul 2022 (E)
(1) 2√2 (2) 2√3
(3) 4√2 (4) 4

Q32*. The equation of a common tangent to the parabolas y = x and y = −(x − 2) is 2 2

26 Jul 2022 (E)
(1) y = 4(x − 2) (2) y = 4(x − 1)
(3) y = 4(x + 1) (4) y = 4(x + 2)

Q33*. The sum of diameters of the circles that touch (i) the parabola 75x 2
= 64(5y − 3) at the point ( 8

5
,
6

5
) and (ii)
the y-axis, is equal to _____ . 25 Jul 2022 (M)

Q34*. The tangents at the points A(1, 3) and B(1, −1) on the parabola y 2
− 2x − 2y = 1 meet at the point P . Then
the area (in unit ) of the triangle P AB is:
2
25 Jul 2022 (E)
(1) 4 (2) 6
(3) 7 (4) 8

Q35^. Let P : y
2
= 4ax, a > 0 be a parabola with focus S .Let the tangents to the parabola P make an angle of π

with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are
collinear is: 29 Jun 2022 (E)
(1) 8 only (2) 2 only
(3) 1

4
only (4) any a > 0

Q36. If vertex of parabola is (2, −1) and equation of its directrix is 4x − 3y = 21, then the length of latus rectum is
28 Jun 2022 (E)
(1) 2 (2) 8
(3) 12 (4) 16

Q37. If the equation of the parabola, whose vertex is at (5, 4) and the directrix is 3x + y − 29 = 0, is
x
2
+ ay
2
+ bxy + cx + dy + k = 0 , then a + b + c + d + k is equal to 27 Jun 2022 (E)
(1) 575 (2) −575
(3) 576 (4) −576

Q38*. A circle of radius 2 unit passes through the vertex and the focus of the parabola y 2
= 2x and touches the
27 Jun 2022 (M)
2
parabola y = (x − 1

4
) + α, where α > 0. Then (4α − 8) is equal to ______.
2

Q39*. Let the common tangents to the curves 4(x 2 2

+ y ) = 9 and y 2
= 4x intersect at the point Q. Let an ellipse,
centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e
and l respectively denote the eccentricity and the length of the latus rectum of this ellipse, then l

e
2
is equal to
______. 26 Jun 2022 (M)

Q40*. Let the normal at the point P on the parabola y 2

= 6x pass through the point (5, −8). If the tangent at P to
the parabola intersects its directrix at the point Q, then the ordinate of the point Q is 26 Jun 2022 (M)
(1) −9

4
(2) 9

(3) −5

2
(4) −3

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Parabola JEE Main 2024 - 2019 PYQs
Chapter Priority: Priority D MathonGo

Q41. Let x = 2t, y = t

3
be a conic. Let S be the focus and B be the point on the axis of the conic such that
SA ⊥ BA , where A is any point on the conic. If k is the ordinate of the centroid of the ΔSAB, then lim k is
t→1

equal to 25 Jun 2022 (M)

(1) 17

18
(2) 19

18

(3) 11

18
(4) 13

18

Q42*. If the line y = 4 + kx, k > 0 , is the tangent to the parabola y = x − x at the point P and V is the vertex of
2

the parabola, then the slope of the line through P and V is 25 Jun 2022 (E)
(1) 3

2
(2) 26

(3) 5

2
(4) 23

Q43*. If y = m 1x + c1 and y = m 2x + c2 , m1 ≠ m2 are two common tangents of circle x 2

+ y
2
= 2 and parabola
y
2
= x , then the value of 8| m 1 m2 | is equal to 25 Jun 2022 (M)
(1) 3√2 − 4 (2) 6√2 − 4
(3) −5 + 6√2 (4) 3 + 4√2

Q44. Let P be a parabola with vertex (3, 2) and focus (4, 4) and P be its mirror image with respect to the line
1 2

x + 2y = 6 . Then the directrix of P is x + 2y = _____.

2 24 Jun 2022 (E)

Q45*. If two tangents drawn from a point (α, β) lying on the ellipse 25x 2
+ 4y
2
= 1 to the parabola y 2
= 4x are
2
such that the slope of one tangent is four times the other, then the value of (10α + 5) 2
+ (16β
2
+ 50) equals
______ 24 Jun 2022 (M)

Q46*. Let x 2
+ y
2
+ Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x at (2, 4) 2

. Then A + C is equal to _____ 24 Jun 2022 (M)

(1) 16 (2) 88

(3) 72 (4) −8

Q47. A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve
C at the point P meets the x-axis at Q. If the y-axis bisects the segment P Q, then C is a parabola with
24 Jun 2022 (E)
(1) length of latus rectum 3 (2) length of latus rectum 6
(3) focus ( 4

3
, 0) (4) focus (0, 3

3
)

Q48*. Consider the parabola with vertex ( 1

2
,
3

4
) and the directrix y = 1

2
. Let P be the point where the parabola
meets the line x = − . If the normal to the parabola at P intersects the parabola again at the point Q. then
1

(PQ)
2
is equal to : 01 Sep 2021 (E)
(1) 25

2
(2) 75

(3) 125

16
(4) 15

Q49. The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R
and S(> R) respectively from the origin, is : 31 Aug 2021 (M)
(1) 2( S − R) (2) 2(S + R)
(3) 4(S − R) (4) 4(S + R)

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Parabola JEE Main 2024 - 2019 PYQs
Chapter Priority: Priority D MathonGo

Q50*. A tangent line L is drawn at the point (2, − 4) on the parabola y 2

= 8x. If the line L is also tangent to the
circle x 2
+ y
2
= a, then a is equal to . 31 Aug 2021 (E)

Q51. If two tangents drawn from a point P to the parabola y 2

= 16(x − 3) are at right angles, then the locus of
point P is: 27 Aug 2021 (E)
(1) x + 4 = 0 (2) x + 2 = 0
(3) x + 3 = 0 (4) x + 1 = 0

Q52*. A tangent and a normal are drawn at the point P (2, −4) on the parabola y 2
= 8x, which meet the directrix of
the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then
2a + b is equal to 27 Aug 2021 (M)
(1) −12 (2) −20
(3) −16 (4) −18

Q53*. If a line along a chord of the circle 4x 2

+ 4y
2
+ 120x + 675 = 0 , passes through the point (−30, 0) and is
tangent to the parabola y 2
= 30x , then the length of this chord is: 26 Aug 2021 (M)
(1) 5 (2) 7
(3) 3√ 5 (4) 5√3

Q54*. Let a parabola P be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from
the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then
the area (in sq. units) of ΔSOR is equal to : 25 Jul 2021 (M)
(1) 16√2 (2) 16
(3) 32 (4) 8√2

Q55*. If the point on the curve y 2

= 6x, nearest to the point (3, 3

2
) is (α, β), then 2(α + β) is equal to _________.
20 Jul 2021 (E)

Q56*. Let y = mx + c, m > 0 be the focal chord of y 2

= −64x , which is tangent to (x + 10) 2
+ y
2
= 4 . Then, the
value of 4√2( m + c) is equal to______ 20 Jul 2021 (M)

Q57*. Let the tangent to the parabola S : y

2
= 2x at the point P (2, 2) meet the x−axis at Q and normal at it meet
the parabola S at the point R. Then the area (in sq. units) of the triangle P QR is equal to: 20 Jul 2021 (M)
(1) 25

2
(2) 35

(3) 15

2
(4) 25

Q58. Let P be a variable point on the parabola y = 4x 2

+ 1. Then, the locus of the mid-point of the point P and the
foot of the perpendicular drawn from the point P to the line y = x is: 20 Jul 2021 (E)
(1) (3x − y) 2
+ (x − 3y) + 2 = 0 (2) 2(3x − y) 2
+ (x − 3y) + 2 = 0

(3) (3x − y) 2
+ 2(x − 3y) + 2 = 0 (4) 2(x − 3y) 2
+ (3x − y) + 2 = 0

Q59*. Let C be the locus of the mirror image of a point on the parabola y 2
= 4x with respect to the line y = x. Then
the equation of tangent to C at P (2, 1) is : 16 Mar 2021 (E)
(1) x − y = 1 (2) 2x + y = 5
(3) x + 3y = 5 (4) x + 2y = 4

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Parabola JEE Main 2024 - 2019 PYQs
Chapter Priority: Priority D MathonGo

Q60*. If the three normals drawn to the parabola, y 2

= 2x pass through the point (a, 0), a ≠ 0 , then a must be
greater than : 16 Mar 2021 (M)
(1) 1

2
(2) − 1

(3) −1 (4) 1

Q61*. A tangent is drawn to the parabola y 2

= 6x which is perpendicular to the line 2x + y = 1. Which of the
following points does NOT lie on it? 25 Feb 2021 (M)
(1) (0, 3) (2) (4, 5)
(3) (5, 4) (4) (−6, 0)

Q62*. A line is a common tangent to the circle (x − 3) 2

+ y
2
= 9 and the parabola y 2
= 4x . If the two points of
contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a + c) is equal to ___ .
25 Feb 2021 (E)

Q63*. If P is a point on the parabola y = x 2

+ 4 which is closest to the straight line y = 4x − 1, then the co-
ordinates of P are: 24 Feb 2021 (E)
(1) (−2, 8) (2) (1, 5)
(3) (2, 8) (4) (3, 13)

Q64. The locus of the mid-point of the line segment joining the focus of the parabola y 2
= 4ax to a moving point of
the parabola, is another parabola whose directrix is: 24 Feb 2021 (M)
(1) x = a (2) x = 0
(3) x = − a

2
(4) x = a

Q65*. The centre of the circle passing through the point (0, 1) and touching the parabola y = x at the point (2, 2
4)

is : 06 Sep 2020 (E)

(1) ( −53

10
,
16

5
) (2) ( 6

5
,
53

10
)

(3) ( 3

10
,
16

5
) (4) ( −16

5
,
53

10
)

Q66*. Let L be a tangent to the parabola y

1
2
= 4(x + 1) and L be a tangent to the parabola y
2
2
= 8(x + 2) such
that L and L intersect at right angles. Then L and L meet on the straight line:
1 2 1 2

06 Sep 2020 (M)

(1) x + 3 = 0 (2) 2x + 1 = 0
(3) x + 2 = 0 (4) x + 2y = 0

Q67*. If the common tangent to the parabolas, y 2

= 4 x and x
2
= 4 y also touches the circle, x 2
+ y
2 2
= c , then c
is equal to : 05 Sep 2020 (M)
(1) 2√ 2
1
(2) √2
1

(3) 1

4
(4) 1

Q68. Let the latus rectum of the parabola y

2
= 4x be the common chord to the circles C1 and C2 each of them
having radius 2√5. Then, the distance between the centres of the circles C and C is : 1 2

03 Sep 2020 (E)

(1) 12 (2) 8
(3) 8√5 (4) 4√5

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Parabola JEE Main 2024 - 2019 PYQs
Chapter Priority: Priority D MathonGo

Q69. Let P be a point on the parabola, y 2

= 12x and N be the foot of the perpendicular drawn from P , on the axis
of the parabola. A line is now drawn through the mid-point M of P N , parallel to its axis which meets the
parabola at Q. If the y−intercept of the line NQ is 4

3
, then : 03 Sep 2020 (M)
(1) P N = 4 (2) M Q = 1

(3) M Q = 1

4
(4) P N = 3

Q70. The area (in sq. units) of an equilateral triangle inscribed in the parabola y 2
= 8x, with one of its vertices on
the vertex of this parabola is 02 Sep 2020 (E)
(1) 64√3 (2) 256√3
(3) 192√3 (4) 128√3

Q71. If one end of a focal chord AB of the parabola y 2

= 8x is at A( 1

2
, −2), then the equation of the tangent to it at
B is: 09 Jan 2020 (E)
(1) 2x + y − 24 = 0 (2) x − 2y + 8 = 0
(3) x + 2y + 8 = 0 (4) 2x − y − 24 = 0

Q72. The locus of a point which divides the line segment joining the point (0, − 1) and a point on the parabola
x
2
= 4y internally in the ratio 1 : 2 is: 08 Jan 2020 (M)
(1) 9x 2
− 12y = 8 (2) 9x 2
− 3y = 2

(3) x 2
− 3y = 2 (4) 4x 2
− 3y = 2

Q73*. Let a line y = mx(m > 0), intersect the parabola, y 2

= x , at a point P , other than the origin. Let the tangent
to it a P , meet the x-axis at the point Q. If area (ΔOP Q) = 4 square unit, then m is equal to
08 Jan 2020 (E)

Q74*. If y = mx + 4 is a tangent to both the parabolas, y 2

= 4x and x 2
= 2by, then b is equal to 07 Jan 2020 (M)
(1) −32 (2) −64
(3) −128 (4) 128

Q75*. The tangents to the curve y = (x − 2) at its points of intersection with the line x − y = 3, intersect at
2
− 1

the point: 12 Apr 2019 (E)

(1) ( 5

2
, 1) (2) ( 5

2
, −1)

(3) (− 5

2
, −1) (4) (− 5

2
, 1)

Q76*. If the line ax + y = c, touches both the curves x 2

+ y
2
= 1 and y 2
= 4√ 2x , then |c| is equal to:
10 Apr 2019 (E)
(1) 1

2
(2) √2
(3) √2
1
(4) 2

Q77. If one end of a focal chord of the parabola, y 2

= 16x is at (1, 4), then the length of this focal chord is
09 Apr 2019 (M)
(1) 24 (2) 25
(3) 22 (4) 20

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Q78*. The area (in sq. units) of the smaller of the two circles that touch the parabola, y 2
= 4x at the point (1, 2)

and the x -axis is 09 Apr 2019 (E)

(1) 8π(3 − 2√2) (2) 8π(2 − √2)
(3) 4π(3 + √2) (4) 4π(2 − √2)

Q79*. The tangent to the parabola y 2

= 4x at the point where it intersects the circle x 2
+ y
2
= 5 in the first
quadrant, passes through the point: 08 Apr 2019 (E)
(1) ( 1

4
,
3

4
) (2) (− 1

3
,
4

3
)

(3) (− 1

4
,
1

2
) (4) ( 3

4
,
7

4
)

Q80. Let P (4, −4) and Q(9, 6) be two points on the parabola, y 2
= 4x and let X be any point on the arc P OQ of
this parabola, where O is the vertex of this parabola, such that the area of ΔP XQ is maximum. Then this
maximum area (in sq. units) is : 12 Jan 2019 (M)
(1) 625

4
(2) 75

(3) 125

4
(4) 125

Q81*. The equation of a tangent to the parabola, x 2

= 8y , which makes an angle θ with the positive direction of x−
axis, is 12 Jan 2019 (E)
(1) y = x tan θ + 2 cot θ (2) y = x tan θ − 2 cot θ
(3) x = y cot θ + 2 tan θ (4) x = y cot θ − 2 tan θ

Q82. If the area of the triangle whose one vertex is at the vertex of the parabola, y 2
+ 4 (x − a ) = 0
2
and the other
two vertices are the points of intersection of the parabola and y -axis, is 250 sq. units, then a value of 'a' is :
11 Jan 2019 (E)
(1) 5√5 (2) 5 (2 1/3
)

(3) (10) 33
(4) 5

Q83*. If the parabolas y 2

= 4b(x − c) and y 2
= 8ax have a common normal, then which one of the following is a
valid choice for the ordered triad (a, b, c) 10 Jan 2019 (M)
(1) (1, 1, 3) (2) ( 1

2
, 2, 0)

(3) ( 1

2
, 2, 3) (4) All of above

Q84. The length of the chord of the parabola x 2

= 4y having equation x − √2y + 4√2 = 0 is 10 Jan 2019 (E)
(1) 6√3 units (2) 8√2 units
(3) 2√11 units (4) 3√2 units

Q85*. Equation of a common tangent to the circle, x 2

+ y
2
− 6x = 0 and the parabola, y 2
= 4x is: 09 Jan 2019 (M)
(1) 2√3y = −x − 12 (2) √3y = x + 3
(3) √3y = 3x + 1 (4) 2√3y = 12x + 1

Q86. Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the
origin, on the positive x-axis then which of following points does not lie on it? 09 Jan 2019 (M)
(1) (6, 4√2) (2) (5, 2√6)
(3) (8, 6) (4) (4, −4)

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Q87. Let A(4, −4) and B(9, 6) be points on the parabola, y 2

= 4x. Let C be chosen on the arc AOB of the
parabola, where O is the origin, such that the area of ΔACB is maximum. Then, the area (in sq. units) of
ΔACB , is: 09 Jan 2019 (E)
(1) 32 (2) 31 3

(3) 30 1

2
(4) 31 1

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Parabola JEE Main 2024 - 2019 PYQs
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ANSWER KEYS
1. (36) 2. (80) 3. (75) 4. (68) 5. (108) 6. (10) 7. (72) 8. (4)
9. (3) 10. (4) 11. (192) 12. (3) 13. (4) 14. (32) 15. (2) 16. (1)
17. (16) 18. (16) 19. (146) 20. (3) 21. (4) 22. (4) 23. (10) 24. (3)
25. (2) 26. (2) 27. (1) 28. (9) 29. (2) 30. (4) 31. (3) 32. (2)
33. (10) 34. (4) 35. (4) 36. (2) 37. (4) 38. (63) 39. (4) 40. (1)
41. (4) 42. (3) 43. (1) 44. (10) 45. (2929) 46. (1) 47. (1) 48. (3)
49. (3) 50. (2) 51. (4) 52. (3) 53. (3) 54. (2) 55. (9) 56. (34)
57. (1) 58. (2) 59. (1) 60. (4) 61. (3) 62. (9) 63. (3) 64. (2)
65. (4) 66. (1) 67. (2) 68. (2) 69. (3) 70. (3) 71. (2) 72. (1)
73. (0.5) 74. (3) 75. (2) 76. (2) 77. (2) 78. (1) 79. (4) 80. (3)
81. (3) 82. (4) 83. (4) 84. (1) 85. (2) 86. (3) 87. (4)

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