MRS CPP
PARABOLA
Subjective Type :
1. (i) Find equation of the parabola with focus (3, -4) and directrix 6x – 7y + 5 = 0
(ii) Identify the conic represented by equation px qy 1, p, q R, p, q 0
(iii) Identify the conic represented by equation 2 xy 4 x 6 y 17 0
2. (i) Find the vertex, axis, focus, directrix, length of latus rectum of the parabola
4y2+12x–20y+67=0
(ii) Find equation of the parabola whose focus is (1, -1) and vertex is (2, 1). Also find
length of latus rectum.
(iii) Find equation of the parabola whose extremities of latus rectum are
(3, 6) and (3, -2).
(iv) Find the equation of the parabola with vertex at origin and satisfying the condition
focus (4, 0) directrix x + 4 = 0.
(v) Find equation of parabola whose focus is (0, 0) and tangent at the vertex is
x–y+1=0
(vi) Find the vertex, axis, focus and latus rectum of the parabola whose parametric
1 2
equations are x u cos t , y u sin t gt where, u, , g
2
are constant t is parameter.
3. (i) A double ordinate of the curve y2 = 4px is of length 8p. Prove that the lines from the
vertex to its two ends are at right angles.
(ii) If y1, y2, y3 be the ordinates of vertices of the triangle inscribed in the parabola y2 = 4ax
1
then show that area of the triangle is y1 y2 y2 y3 y3 y1
8a
(iii) Find the interval of a for which the point (-2a, a + 1) lies in the smaller region bounded
by the circle x2 + y2 = 4 and the parabola y2 = 4x
(iv) If PQ be a focal chord and S be the focus of a parabola then prove that Sp, 2a, SQ
are in H.P. (where length of latus rectum = 4a)
4. (i) A tangent to the parabola y2 = 4ax make an angle of 60o with x axis, find its point
of contact.
(ii) Prove that the straight line x + y = 1 touches the parabola y = x – x2.
(iii) Prove that the line lx my n 0 touches the parabola y 2 4a x b
2 2
if am bl nl
5. Let x – y + 1 = 0 be the equation of tangent at any point (2, 3) on the parabola whose focus
is (-1, 1). Find the length of latus rectum of the parabola.
a
6. (i) Prove that the straight line y = mx + c touches the parabola y 2 = 4a (x + a) if c = am + .
m
(ii) If from the vertex of the parabola y2 = 4ax a pair of chords be drawn perpendicular to
each other and with these chords as adjacent sides a rectangle is completed then
prove that the locus of the further vertex of the rectangle is y2 = 4a(x – 8a).
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MRS CPP
(iii) Tangents to the parabola y2 = 4ax are drawn at points whose abscissa are in the ratio
m2 : 1 prove that the locus of their point of intersection is the curve
y 2 m1 / 2 m 1 / 2 ax . 2
(iv) Equation of tangents drawn at P, Q and vertex A of a parabola are 3x + 4y – 7 = 0,
2x + 3y – 1 = 0 and x – y = 0 respectively. Find co-ordinates of the locus and
length of latus rectum of the parabola.
7. Prove that the straight lines, one a tangent to the parabola y 2 = 4a(x + 1) and other to the
parabola y2 = 4a'(x + a') which are at right angles to one another, meet on the straight line
x + a + a' = 0. Show also that this straight line is the common chord of the two parabolas.
8. Through the vertex O of a parabola y2 = 4x chords OP and OQ are drawn at right angles to
one another show that for all positions of P, PQ cuts the axis of the parabola at a fixed
point. Also find locus of the middle point of PQ.
9. Let P be the point (2a, 0) and QR be a variable chord of the parabola y2 = 4ax passing
1 1 1
through P. Prove that = constant = .
PQ 2 PR 2 4a 2
10. Find the equation of common tangents to
a) y2 = 4ax and (x + a)2 + y2 = a2
b) y2 = 4x and x2 = 4y
11. (i) Prove that the length of intercept on the normal at P(at2, 2at) of the parabola y2 = 4ax
made by the circle described on the line joining the focus and P as diameter is
a 1 t2
.
(ii) If the normal to the parabola y2 = 4ax makes an angle with the axis of the parabola,
1 tan
show that it will cut the curve again at angle tan
2
(iii) If two chords are drawn from the point (4, 4) to the parabola x 2 = 4y are divided by the
line y = mx in the ratio 1 : 2 then find the interval in which m lies
12. A parabola is drawn to pass through A and B the ends of a diameter of a given circle of
radius ‘a’ and to have as directrix a tangent to a concentric, circle of radius ‘b’ the axes of
reference being AB and a perpendicular diameter. Prove that the locus of the focus of the
x2 y2
parabola is 1.
b2 b2 a2
13. (i) If P, Q, R be three co-normal points of the parabola y2 = 4ax, whose normal pass through
the point T then prove that SP. SQ. SR = a(ST)2.
(ii) Find the shortest distance between the parabola y 2 4 x and the circle
x 2 y 2 6 x 12 y 20 0
(iii) Find the minimum value of the expression
z cos tan 2 3 sin 2 tan 2 where and are independent parameters.
2
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MRS CPP
a2
14. From a point A common tangents are drawn to the circle x2 + y2 = and parabola y2 =
2
4ax. Find the area of the quadrilateral formed by the common tangents, the chord of
contacts of the circle and that of the parabola.
15. Find minimum distance between the curves y2 = x-1 and x2 = y-1.
Objective Type :
1. two different tangents of y2 = 4x are the normals to x2 = 4by then
1 1 1 1
(a) b (b) b (c) b (d) b
2 2 2 2 2 2
1
2. A focal chord of y 2 4ax meets in P and Q. If S is the focus, then is equal to
SP SQ
1 2 4
(a) (b) (c) (d) none of these
a a a
3. Let the line lx my 1 cut the parabola y 2 4ax in the points A and B. Normals at A and B meet
at point C. Normal from C other than these two meet the parabola at D, then the coordinate of D are
4am 4a 2am 2a 4am 2 4am
(a) a,2a (b) 2 , (c) 2 , (d) 2 ,
l l l l l l
4. 2 2
The length of the latus-rectum of the parabola 169 x 1 y 3 5 x 12 y 17 is
2
14 12 28
(a) (b) (c) (d) none of these
13 13 13
5. A double ordinate of the parabola y 2 8 px is of length 16 p. The angle subtended by it at the vertex
of the parabola is
(a) (b) (c) (d)
4 2 3
6. The normals at three points P, Q, R of the parabola y 2 4ax meet in h, k . The centroid of triangle
PQR lies on
(a) x 0 (b) y 0 (c) x a (d) y a
7. The condition that the parabolas y 2 4cx d and y 2 4ax have a common normal other than x-
axis a c 0 is
(a) 2a 2c d (b) 2c 2a d (c) 2d 2a c (d) 2d 2c a
8. A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is
y 22 4x 1. After reflection, the ray must pass through the point
(a) 2,0 (b) 1,2 (c) 0,2 (d) 2,0
9. The mirror image of the directrix of the parabola y 2 4 x 1 in the line mirror x 2 y 3 is
(a) x 2 (b) 4 y 3 x 16 (c) x 3 y 0 (d) x y 0
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MRS CPP
10. The shortest distance between the parabolas y 2 4 x and y 2 2 x 6 is
(a) 2 (b) 5 (c) 3 (d) none of these
11.
If the normal to the parabola y 2 4ax at the point at 2 ,2at cuts the parabola again at aT 2 ,2aT ,
then
(a) 2 T 2 (b) T ,8 8,
(c) T 2 8 (d) T 2 8
12. The length of latus- rectum of the parabola whose parametric equation is
x t 2 t 1, y t 2 t 1, where t R is equal to
(a) 2 (b) 2 (c) 2 2 (d) 4
13. The line x 1 0 is the directrix of the parabola y 2 kx 8 0. Then one the values of k is
1 1
(a) (b) 8 (c) 4 (d)
8 4
14. If the tangent and normal at any point P of a parabola meet the axes in T and G respectively, then
(a) ST SG SP (b) ST SG SP (c) ST SG SP (d) ST SG.SP
15. On the parabola y x 2 , the point least distance from the straight line y 2 x 4 is
(a) 1,1 (b) 1,0 (c) 1,1 (d) 0,0
The equation of the common tangent touching the circle x 3 y 2 9 and the parabola y 2 4 x
2
16.
above the x-axis, is
(a) 3 y 3 x 1 (b) 3 y x 3 (c) 3 y x 3 (d) 3 y 3 x 1
17. The equation of latus rectum of a parabola is x y 8 and the equation of the tangent at the vertex
is x y 12, then length of the latus rectum is
(a) 4 2 (b) 2 2 (c) 8 (d) 8 2
18. The equation of the parabola whose focus is the point 0,0 and the tangent at the vertex is
x y 1 0 is
(a) x 2 y 2 2 xy 4 x 4 y 4 0 (b) x 2 y 2 2 xy 4 x 4 y 4 0
(c) x 2 y 2 2 xy 4 x 4 y 4 0 (d) x 2 y 2 2 xy 4 x 4 y 4 0
19. Two common tangents to the circle x 2 y 2 2a 2 and parabola y 2 8ax are
(a) x y 2a (b) y x 2a (c) x y a (d) y x a
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MRS CPP
20. The locus of the midpoint of the line segment joining the focus to a moving point on the parabola
y 2 4ax is another parabola with the directrix
a a
(a) x a (b) x (c) x 0 (d) x
2 2
21. If a 0 and the line 2bx 3cy 4d 0 passes through the points of intersection of the parabolas
y 2 4ax and x 4ay, then
(a) d 2 3b 2c (b) d 2 3b 2c 0
2 2
(c) d 2 2b 3c 0 (d) d 2 2b 3c 0
2 2
22. The point on parabola 2 y x 2 , which is nearest to the point 0,3 is
(a) 4,8 (b) 1,1 / 2 (c) 2,2 (d) none of these
23. The normal meet the parabola y 2 4ax at that point where the abscissae of the point is equal to the
ordinate of the point is
(a) 6a,9a (b) 9a,6a (c) 6a,9a (d) 9a,6a
24. Tangent to the parabola y x 2 6 at 1,7 touches the circle x 2 y 2 16 x 12 y c 0 at the point
(a) 6,7 (–13,–9) (c) 6,7 (d) 13,7
The equations of the common tangents to the parabolas y x 2 and y x 2 is/ are
2
25.
(a) y 4x 1 (b) y 1 (c) y 4 x 1 (d) y 3 x 50
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Ranchi (SOP, Doranda): Samriddhi Complex, Ground Floor, Near St. Xavier’s School, South Office Para Doranda, Ranchi.
Corp. off. : FIITJEE House 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi – 16. Ph : 46106000/10/13/15, Toll Free No. 1800114242.
Fax : 011-26513942, Web:www.fiitjee.com
MRS CPP
Answers
(Parabola)
Subjective Type:
1. (7x + 6y)2 – 570x + 750y + 2100 = 0
y 2 8 x 5 ; y 2 8 x 1 ,
2 2
2. (ii) 4x2 + y2 – 4xy + 8x + 46y – 71 = 0, (iii)
(iv) y2 = 16x(v) x2 + y2 + 2xy – 4x + 4y – 4 = 0,
u 2 sin 2 u 2 sin 2
2 2
(vi) Vertex 2 g ,
2g latus rectum 2u cos / g
3. (iii) a 1, 5 2 6 5. 2
13
8. y2 = 2(x – 4)
10. a) 2 y x 3a b) x+y+1=0 11. (i) y(y2 + (3a – x)a) = 0,
(iii) m , 3 1 3 1,
13. (ii) 4 2 5 (iii) 3 2 2
15a 2 3 2
14. . 15. .
4 4
Objective Type :
1. (b) 2. (a) 3. (d) 4. (c)
5. (b) 6. (b) 7. (a) 8. (c)
9. (b) 10. (b) 11. (d) 12. (a)
13. (c) 14. (c) 15. (a) 16. (c)
17. (d) 18. (c) 19. (b) 20. (c)
21. (d) 22. (c) 23. (d) 24. (c)
25. (a)
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