CONIC SECTION
CONIC SECTION
Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point(s) of tangency.
INITIAL STEP EXERCISE
1. The equation of the common tangent touching the circle (x – 3)2 + y2 = 9 and the parabola y2 = 4x above the x-axis is (a) (b) (c) (d) 2. √3 y = 3x + 1 √3 y = –(x + 3) √3 y = x + 3 √3 y = – (3x + 1)
The triangle formed by the tangent to the curve f(x) = x2 + bx – b at the point (1, 1) and the coordinates axes lies in the first quadrant. If its area is 2, then the value of b is (a) (c) –1 –3 1/8 4 3 –9 (b) (d) (b) (d) (b) (d) 3 1 8 1/4 9 –3
3.
If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the values of k is (a) (c)
4.
If x + y = k is normal to y2 = 12 x, then k is (a) (c)
5.
y – 2x – 2y + 5 = 0 is
2
(a) (b) (c) (d) 6.
a circle with centre (1, 1) a parabola with vertex (1, 2)
x=
a parabola with directrix a parabola with directrix
3 2 1 2
x=−
For the ellipse x2 + 4y2 = 9 (a) (b)
1 the eccentricity is 2 3 the latus rectum is 2
(c) (d) 7.
a focus is (3√3, 0) a directrix is x = 2√3
8.
9.
x2 + y2 = 1 The number of values of c such that the straight line y = 4x + c touches the curve 4 is (a) 0 (b) 1 (c) 2 (d) Infinite 2 The tangents to the hyperbola x – y2 = 3 are parallel to the straight line 2x + y + 8 = 0 at the following points (a) (2, 1) (b) (2, –1) (c) (–2, 1) (d) (–2, –1) Let P (a sec q, b tan q) and Q (a sec f, b tan f), where q + f = p/2 be two points on the hyperbola x 2 y2 − =1 a 2 b2 . If (h, k) is the point of intersection of the normals at P and Q, then k is equal to
a 2 + b2 a
(a)
(b)
2
⎛ a 2 + b2 ⎞ −⎜ ⎟ ⎜ a ⎟ ⎠ ⎝ ⎛ a 2 + b2 ⎞ ⎜ ⎟ ⎜ b ⎟ ⎠ ⎝
(c) 10. (a) (c)
⎛a +b ⎜ ⎜ b2 ⎝
2
⎞ ⎟ ⎟ ⎠
(d) (b) (d)
If the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 cut the co-ordinates axes is concylic points, then a1a2 = b1b2 a1b2 = a2b1 a1b1 = a2b2 none of these
11.
x 2 y2 x 2 y2 − 2 =1 − 2 =2 2 2 b b , tangents are drawn to the hyperbola a . The From any point on the hyperbola a area cut off by the chord of contact on the asymptotes is equal to
(a) (c)
ab 2
2ab
(b) (d)
ab 4ab
12.
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half of the distance between foci is (a)
2 3
(b) (b) (d)
2 2
4 3
√2 none of these
(c)
4 3
(d)
none of these
13.
The eccentricity of the rectangular hyperbola is (a) (c) 2 0
14.
x y + =1 represents : The equation 12 − k 8 − k
(a) (b) (c) (d) a hyperbola if k < 8 an ellipse is k > 8 a hyperbola if 8 < k < 12 none of the above
15.
A common tangent to 9x2 – 16y2 = 144 and x2 + y2 = 9 is
y=
(a)
3 15 x+ 7 7
y=3
(b)
y=2
2 15 x+ 7 7
3 x + 15 7 7
(c) (d) 16.
none of the above
If y = m x + c touches the parabola y2 = 4a (x + a), then
c=
(a) 17. (a) (c) 18.
a m
c = am +
(b) (b) (d) (0, –1) (1, 3)
a m
c=a+
(c)
a m
2
(d)
none of these
The co-ordinates of an end-point of the latus rectum of the parabola (y – 1) = 4(x + 1) are (0, –3) (0, 1)
The latus rectum of a parabola whose focal chord is P S Q such that S P = 3 and S Q = 2 is given by (a)
24 5
2 4
(b) (b) (d)
12 5
1 none of these
(c)
6 5
(d)
none of these
19.
The length of the latus rectum of the parabola whose focus is (3, 3) and directrix is 3x – 4y – 2 = 0 is (a) (c)
20.
An equilateral triangle is inscribed in the parabola y2 = 4ax whose one vertex is at the vertex of the parabola. Then length of its side is (a) (c) 4a √3 16a √3 (b) (d) 2a √3 8a √3
21.
If the segment intercepted by the parabola y2 = 4ax with the line lx + my + n = 0 subtends a right angle at the vertex, then (a) (b) (c) (d) 4al + n = 0 4al + 4am + n = 0 4am + n = 0 al + n = 0
22.
Three normals to the parabola y2 = x are drawn through a point (c, 0) then
c=
(a) 23.
1 4
c=
(b)
1 2
2
c>
(c)
1 2
(d)
none of these
The slope of a chord of the parabola y = 4ax which is normal at one end and which subtends a right angle at (a) (c) 2
1 2
(b) (d)
2
none of these
24.
If the normal at (1, 2) on the parabola y2 = 4ax meets the parabola again at the point (t2, 2t), then the value of t is (a) (c) 1 –3 (b) (d) 3 1
25.
x 2 y2 + =1 2 The eccentric angle of a point on the ellipse 6 whose distance from the centre of the ellipse is 2 is
(a) (c) π/4 5π/3
2 2
(b) (d)
3π/2 7π/6
26.
x y + =1 10 − a 4 − a The equation represents an ellipse if
(a) (c) a<4 4 < a < 10 (b) (d) a>4 a > 10
27.
x 2 y2 + 2 =1 2 b included between The locus of the middle point of the portion of the tangent of the ellipse a the axes is the curve
(a)
x 2 y2 + =4 a 2 b2
(b)
x 2 y2 − =4 a 2 b2
(c)
x 2 b2 + =4 a 2 y2
(d)
none of these
28.
x 2 b2 + 2 =1 2 y are at If chords of contact of tangents from two points (x1, y1) and (x2, y2) to the ellipse a x1x 2 yy right angles, then 1 2 is equal to
(a) (c) 29.
a2 b2 − a b4
4
−
(b)
b2 a2 b4 a4
−
(d)
x 2 y2 + 2 =1 2 b (a < b) and P(x1, y1) is a point on it, then SP + S′P is If S and are two foci of an ellipse a equal to (a) 2a (b) 2b (d) b + e y1 (c) a + e x1
x 2 y2 + 2 =1 2 b , if The line lx + my + n = 0 is a normal to the ellipse a
(a) (b) (c) (d)
30.
a 2 b 2 (a 2 − b 2 ) 2 + = m 2 l2 n2 a 2 b 2 (a 2 − b 2 ) 2 + = l2 m 2 n2
a 2 b 2 (a 2 − b 2 ) 2 − = l2 m 2 n2
none of the above
31.
If the focal distance of an end of the minor axis of an ellipse (referred to its axes as the axes of x and y respectively) is k and the distance between its foci is 2h, then its equation is (a)
x 2 y2 + =1 k2 h2
(b) (c) (d) 32.
x2 y2 + 2 =1 k2 k − h2 x2 y2 + 2 =1 2 k h − k2 x2 y2 + 2 =1 k2 k + h2
x y + =1 The eccentricity of the ellipse which meets the straight line 7 2 on the axis of x and the straight x y − =1 line 3 5 on the axis of y and whose axes lie along the axes of co-ordinates is :
3 2 7 3 7 2 6 7
2
(a) 33.
(b)
(c)
(d)
2
none of these
x y α β + 2 =1 tan tan 2 b 2 2 is , then If α and β are eccentric angles of the ends of a focal chord of the ellipse a equal to
(a)
1− e 1+ e
(b)
e −1 e +1
(c)
e +1 e −1
(d)
none of these
34.
The equation of the parabola whose vertex the focus lie on the axis of x at distances a and a1 from the origin respectively is (a) (b) (c) (d) y2 = 4(a1 – a) x y2 = 4(a1 – a) (x – a) y2 = 4 (a1 – a) (x – a1) None of these
35.
x2 y2 + =1 represents an ellipse if The equation 10 − a 4 − a
(a) (c) a<4 4 < a < 10 16x – 75y = 418 75x – 16y = 418 25x – 4y = 400 None of these xy = c2 in four points P(x1, y1), Q(x2, y2), R (x3, (b) (d) a>4 a > 10
36.
Equation of the chord of the hyperbola 25x2 – 16y2 = 400 which is bisected at the point (6, 2) is (a) (b) (c) (d)
37.
If the circle x2 + y2 = a2 intersect the hyperbola y3), S(x4, y4), then (a) (b) (c) (d) x1 + x2 + x3 + x4 = 0 y1 + y2 + y2 + y4 = 0 x1 x2 x3 x4 = c4 All the above
38.
A point on the ellipse 4x2 + 9y2 = 36 where the tangent is equally inclined to the axes is
(a)
4 ⎞ ⎛ 9 , ⎜ ⎟ ⎝ 13 13 ⎠ 4 ⎞ ⎛ 9 , ⎜− ⎟ ⎝ 13 13 ⎠ 4 ⎞ ⎛ 9 ,− ⎜ ⎟ 13 ⎠ ⎝ 13
all the above
(b)
(c) (d) 39.
40.
41.
42.
43.
x 2 + ( y − 3) 2 + x 2 + ( y + 3) 2 = 6 represents (a) a circle (b) a line segment (c) a pair of lines (d) none of these x = sin2t, y = 2 sin t are the parametric equations of (a) a parabola (b) a portion of a parabola (c) a hyperbola (d) none of these The vertex of the parabola x2 + 2y – 8x + 7 = 0 is (a) (4, 11/2) (b) (4, 9/2) (c) (9/2, 4) (d) none of these The number of point(s) (x, y) (where x and y both are perfect squares of integers) on the parabola y2 = px, p being a prime number, is (a) one (b) zero (c) two (d) infinite If two distinct chords of a parabola y2 = 4ax, passing through (a, 2a) are bisected on the line x + y = 1, then length of the latus-rectum can be (a) 6 (b) 1 (c) 4 (d) 5
The equation
44.
x 2 y2 + 2 =1 2 b , it being given that the length of its latus rectum is half of its The eccentricity of the ellipse a major axis is
(a) (c) 1/2 √2 (b) (d) 1/√2 none of these
45.
The radius of the circle passing through the foci of the ellipse x2/16 + y2/9 = 1, and having its centre at (0, 3) is (a) (c) 4 √(1/2) (–at2, –2at) (–at2, 2at) (a sin2t, – 2a sin t) none of these (b) (d) 3 7/2
46.
Parametric coordinates of any point on the parabola y2 = 4ax can be (a) (b) (c) (d)
47.
For all real values of t, which of the following points lies on the parabola y2 = 4ax
(a) (b) (c) (d) 48.
(–at2, –2at) (–at2, 2at) (a sin2t, – 2a sin t) none of these
If the chord of contact of tangents drawn from a point P to the parabola y2 = 4ax touches the parabola x2 = 4by, then the locus of P is (a) (c) a circle a parabola (b) (d) an ellipse a hyperbola
49.
Length of chord of parabola x2 = 4ay passing through vertex and having slope tan α is (a) (b) (c) (d) 4|a| cosec α cot α 4|a| sec α tan α 4|a| cos α cot α 4|a| sin α tan α
50.
If M is the foot of perpendicular drawn from a point P on a parabola y2 = 4ax to its directrix and SPM is an equilateral triangle, where S is the focus, then SP is equal to (a) (c) a 3a (b) (d)
2
2a 4a
51.
The condition that the parabola y = 4c (x – d) and y2 = 4ax have a common normal other than x-axis (a > 0, c > 0) is (a) (c) 2a < 2c + d 2a < 2c – d (b) (d) 2a > 2c + d 2a > 2c – d
52.
x 2 y2 − =1 from where two perpendicular tangents can be Number of point (s) outside the hyperbola 25 36 drawn to the hyperbola is
(a) (c) zero 2 (b) (d) 1 3
53.
x 2 y2 − 2 =1 2 b such that OPQ is an equilateral triangle, O If PQ is a double ordinate of the hyperbola a being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies
1< e <
(a) (c) 54.
2 3
e=
(b)
2 3
2 3
e = 3/2
e>
(d)
The locus of the point of intersection of normals at the point on the parabola where tangents drawn meet at the directrix is (a) (c) a parabola an ellipse (b) (d) a circle a hyperbola
55.
A point moves in a plane so that its distance PA and PB from two fixed points A and B in the plane satisfy the relation: PA – PB = k (k ≠ 0), then the locus of P is (a) (b) (c) a parabola an ellipse a hyperbola
(d)
a branch of a hyperbola
Solutions:
1. 7. 13. 19. 25. 31. 37. 43. 49. 55.
c c b a a b d b b c
2. 8. 14. 20. 26. 32. 38. 44. 50.
c b c d a c a, d b d
3. 9. 15. 21. 27. 33. 39. 45. 51.
c d b a a b b a a
4. 10. 16. 22. 28. 34. 40. 46. 52.
b a b c a b b d a
5. 11. 17. 23. 29. 35. 41. 47. 53.
c d b b b a b c d
6. 12. 18. 24. 30. 36. 42. 48. 54.
d a a c b b b d a
FINAL STEP EXERCISE
1. The curve described parametrically by x = t2 + t + 1, y = t2 – t + 1 represents: (a) (b) (c) (d) 2. a pair of straight lines an ellipse a parabola a hyperbola
Consider a circle with its centre lying on the focus of the parabola y2 = 2p x such that it touches the directrix of the parabola. Then, a point of intersection of the circle and the parabola is
3.
⎛p ⎞ ⎛ p ⎞ ⎜ ,−p ⎟ ⎜− ,p⎟ 2 ⎝ ⎠ (c) ⎝ 2 ⎠ (b) (d) (a) 2 Three normals to the parabola y = x drawn through a point (c, 0), then
⎛p ⎞ ⎜ , p⎟ ⎝2 ⎠
⎛ p ⎞ ⎜− ,− p⎟ 2 ⎝ ⎠
c=
(a)
1 4 1 2
2 2
(b)
1 2
c>
(c) 4.
(d)
none of these
On the ellipse 4x + 9y = 1, the points at which the tangents are parallel to the line 8x = 9y are
5.
⎛ 2 1⎞ ⎛ 2 1⎞ ⎛ 2 1⎞ ⎛2 1⎞ ⎜ , ⎟ ⎜− , ⎟ ⎜ − ,− ⎟ ⎜ ,− ⎟ ⎝ 5 5 ⎠ (b) ⎝ 5 5⎠ ⎝ 5 5⎠ ⎝ 5 5⎠ (c) (d) (a) If P(x, y), F1 = (3, 0), F2 = (–3, 0) and 16x2 + 25y2 = 400, then PF1 + PF2 equals
(a) (c) 8 10 (b) (d) 6 12
6.
x 2 y2 + =1 The radius of the circle passing through the foci of the ellipse 16 9 and having its centre (0, 3) is
(a) (c) 4 (b) (d)
2 2
3 7/2
√12
7.
8.
x y + =1 4 and C be the circle x2 + y2 = 9. Let P and Q be the points (1, 2) and (2, Let E be the ellipse 9 1) respectively. Then: (a) Q lies inside C but outside E (b) Q lies outside both C and E (c) P lies inside both C and E (d) P lies inside C but outside E If x = 9 is the chord of contact of the hyperbola x2 – y2 = 9, then the equation of the corresponding pair of tangents is (a) 9x2 – 8y2 + 18x – 9 = 0 (b) 9x2 – 8y2 – 18x + 9 = 0 (c) 9x2 – 8y2 – 18x – 9 = 0 (d) 9x2 – 8y2 + 18x + 9 = 0
⎛1 ⎞ 1 P ⎜ , 1⎟ An ellipse has eccentricity 2 and one focus at the point ⎝ 2 ⎠ . Its one directrix is the common tangent, nearer to the point P to the circle x2 + y2 = 1 and the hyperbola x2 – y2 = 1. The equation of the ellipse in standard form is
9.
(a)
1⎞ ⎛ 9⎜ x − ⎟ + 12( y − 1) 2 = 1 3⎠ ⎝ 1⎞ ⎛ 9⎜ x + ⎟ + 12( y − 1) 2 = 1 3⎠ ⎝ 1⎞ ⎛ 9⎜ x + ⎟ + 12( y − 1) 2 = 1 3⎠ ⎝ 1⎞ ⎛ 9⎜ x + ⎟ + 12( y + 1) 2 = 1 3⎠ ⎝
2 2 2
2
(b)
(c)
(d) 10.
If C is the centre and A and B are the points on the conic 4x2 – 9y2 – 8x – 36 y + 4 = 0 such that π ˆ ACB = 2 , then CA–2 + CB–2 is equal to (a) (b) (c) (d)
5 36 13 36 15 36 16 36
11.
If the normals from any point to the parabola x2 = 4y cuts the line y = 2 in points whose abscissae are in A.P., then the slopes of the tangents at 3 co-normal points are in (a) (c) A.P. H.P. (b) (d) G.P. none of these
12.
The area of the rectangle formed by the perpendiculars from the centre of the ellipse to the tangents π and normals at the point whose eccentric angle 4 is
(a)
⎛ a 2 − b2 ⎞ ⎜ 2 ⎟ ⎜ a + b 2 ⎟ab ⎠ ⎝ ⎛ a 2 + b2 ⎞ ⎜ 2 ⎟ ⎜ a − b 2 ⎟ab ⎠ ⎝ 1 ⎛ a 2 − b2 ⎞ ⎟ab ⎜ ab ⎜ a 2 + b 2 ⎟ ⎠ ⎝ 1 ⎛ a 2 + b2 ⎞ ⎟ab ⎜ ab ⎜ a 2 − b 2 ⎟ ⎠ ⎝
(b)
(c)
(d) 13.
x 2 y2 + 2 =1 2 b meets the auxiliary circle in two The tangent at a point P (a cos θ, b sin θ) on the ellipse a points. The chord joining them subtends a right angle at the centre. Then the eccentricity of the ellipse is given by :
(a) (c) (1 + sin2θ)–1/2 (1 + sin2θ)1/2 (b) (d) (1 + cos2θ)–1/2 (1 + cos2θ)1/2
14.
⎛ c⎞ ⎜ ct , ⎟ If the normal at ⎝ t ⎠ on the curve xy = c2 meets the curve again in ' t′' then
15.
1 1 1 1 t′ = − t′ = 2 t ′2 = − 2 t 3 (b) t (c) t (d) t (a) The equation of the chord joining two points (x1, y1) and (x2, y2) on the rectangular hyperbola xy = c2 is t′ = −
x y + =1 x1 + x 2 y1 + y 2 x y + =1 x 1 − x 2 y1 − y 2 x y + =1 y1 + y 2 x1 + x 2 x y + =1 y1 − y 2 x 1 − x 2
(a)
(b)
(c)
(d) 16. (a) (c) 17. (a)
If e1 and e2 are the eccentricities of the conic sections 16x2 + 9y2 = 144 and 9x2 – 16y2 = 144, then : e12 + e22 = 3 e12 + e22 < 3 36 (b) (d) (b) e12 + e22 > 3 e12 – e22 = 1 9x2 – 5y2 = 45, then the value of λ is 6
If the line y = 3x + λ touches the hyperbola
(c) 18. (a) (b) (c) (d) 19.
15
(d)
45
The combined equation of the asymptotes of the hyperbola 2x2 + 5xy + 2y2 + 4x + 5y = 0 is 2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0 2x2 + 5xy + 2y2 + 4x + 5y – 2 = 0 2x2 + 5xy + 2y2 = 0 none of the above
The tangents from (1, 2√2) to the hyperbola 16x2 – 25y2 = 400 include between them an angle equal to (a)
π 6
(b)
π 3
(c)
π 2
(d)
π 4
20.
If PN is the perpendicular from a point on a rectangular hyperbola to its asymptotes, the locus of the mid-point of PN is (a) (c) a circle ellipse (b) (d) parabola hyperbola
21.
x 2 y2 − 2 =1 2 b if The line x cos a + y sin a = p touches the hyperbola a
(a) (b) (c) (d) a2 cos2α – b2sin2α = p2 a2 cos2α – b2sin2α = p a2 cos2α + b2sin2α = p2 a2 cos2α + b2sin2α = p a parabola with focus S at (2, 1) a parabola with vertex at (2, 1) an ellipse with centre at (2, 1) none of the above 4a cosec α cot α (b) 4a cos α cot α
2
22.
The parametric representation (2 + t2, 2t + 1) represents (a) (b) (c) (d)
23.
The length of the chord of the parabola x2 = 4ay passing through the vertex and having slope tan α is (a) (c) 4a tan α sec α 4a sin α tan α
2
(d)
24.
The two parabolas y = 4x and x = 4y intersect at a point P, whose abscissa is not zero, such that (a) (b) (c) (d) they both touches each other at P they cut at right angles at P the tangents to each other at P make complementary angles with the x-axis none of the above
25.
It (at2, 2at) are the co-ordinates of one end of a focal chord of the parabola y2 = 4ax, then the coordinates of the other end are (a) (at2, – 2at) (b) (–at2, –2at)
(c) 26. 27. (a)
⎛ a 2a ⎞ ⎜ 2, ⎟ ⎝t t ⎠
6
2
(d) (b)
⎛ a 2a ⎞ ⎜ 2 ,− ⎟ t ⎠ ⎝t
4 (c) 3
0
If P S Q is the focal chord of the parabola y2 = 8 x such that S P = 6. Then length S Q is (d) none of these The point on the curve y = ax, the tangent at which makes an angle of 45 with x-axis will be given by
28.
⎛a a⎞ ⎛ a a⎞ ⎛a a⎞ ⎛ a a⎞ ⎜ , ⎟ ⎜− , ⎟ ⎜ , ⎟ ⎜− , ⎟ ⎝ 2 4 ⎠ (b) ⎝ 2 4⎠ ⎝ 4 2 ⎠ (d) ⎝ 4 2⎠ (a) (c) The number of point (s) (x, y) (where x and y both are perfect squares of integers) on the parabola y2 = px, p being a prime number is (a) one (b) zero
(c) two (d) infinite
29.
x 2 y2 + 2 =1 2 b with A A′ as the major axis, then the maximum value of P is a variable point on the ellipse a A′ is the area of the triangle A P (a) ab (b) 2ab (c) ab/2 (d) none of these
x 2 y2 + 2 =1 2 b and a concentric circle of radius r is The slope of a common tangent to the ellipse a
30.
tan −1
(a)
r 2 − b2 a2 − r2
(b)
r 2 + b2 a2 + r2
⎛ r 2 − b2 ⎞ ⎜ 2 2⎟ ⎜a −r ⎟ ⎠ ⎝
(c)
(d) 31.
a2 − r2 r 2 − b2
The equation of the tangent to the ellipse 4x2 + 3y2 = 5 which are parallel to the line y = 3x + 7 are
y = 3x ± 155 3 155 12 95 12
(a)
y = 3x ±
(b)
y = 3x ±
(c) (d) 32.
none of these
x 2 y2 + 2 =1 2 b in real points if The line x = at2, meets the ellipse a
(a) (c) |t| < 2 |t| ≤ 1 (b) (d) |t| > 1 none of these
33.
If α and β are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is (a) (c)
cos α + cos β cos(α − β)
cos α − cos β cos(α − β)
(b) (d)
sin α − sin β sin(α − β)
sin α + sin β sin(α + β)
34.
x 2 y2 + 2 =1 2 b If the normal at any point P on the ellipse a meets the axis in G and g respectively, then P G :Pg=
(a) (c) a:b b2 : a 2 (b) (d) a 2 : b2 b:a
35.
If x1, x2, x3 as well as y1, y2, y3 are in G.P. with the same common ratio or in A.P. with the same common difference then the points (x1, y1), (x2, y2) and (x3, y3). (a) (b) (c) (d) lie on a straight line lie on an ellipse lie on a circle are vertices of a triangle
36.
If β1 and β2 are the ordinates of two points A and B on the parabola and β3 is the ordinate of the point of intersection of tangents at A and B, then (a) (b) (c) (d)
β1, β2, β3 are in A.P. β1, β2, β3 are in G.P. β1, β2, β3 are in H.P.
none of these
37.
The centre of a circle passing through the point (0, 1) and touching the curve y = x2 at (2, 4) is
(a)
⎛ − 16 27 ⎞ , ⎟ ⎜ ⎝ 5 10 ⎠ ⎛ − 16 53 ⎞ , ⎟ ⎜ ⎝ 5 10 ⎠
2
(b)
⎛ − 16 5 ⎞ , ⎟ ⎜ ⎝ 7 10 ⎠
(c) 38.
(d)
2
None of these
x y x 2 y2 1 + 2 =1 − = and the hyperbola 144 81 25 coincide. Then the value of b2 is The foci of the ellipse 16 b
(a) (c) 5 9 (b) (d)
2
7 1
39.
The normal at the point (bt1 , 2bt1) on a parabola meets the parabola again in the point (by22, 2bt2), then
t 2 = − t1 −
(a)
2 t1
t 2 = t1 −
(b)
2 t1 2 t1
t 2 = t1 +
(c)
2 t1
t 2 = − t1 +
(d)
Solutions:
1. 7. 13. 19.
c d a c
2. 8. 14. 20.
a b a b
3. 9. 15. 21.
c a a d
4. 10. 16. 22.
b a c a
5. 11. 17. 23.
c b b b
6. 12. 18. 24.
a a a b
25. 31. 37.
d b c
26. 32. 38.
c c b
27. 33. 39.
c d d
28. 34.
b c
29. 35.
a a
30. 36.
a d
ANALYSIS
1. If a ≠ 0 and the line 2bx + 3cy + 4d = 0 passes through the point of intersection of the parabola y2 = 4ax and x2 = 4ay, then (a) (b) (c) [Ans. : c] 2. The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directrices is x = 4, then the equation of the ellipse is (a) (c) [Ans. : b] 3. 4x2 + 3y2 = 12 3x2 + 4y2 = 1 (b) (d) 3x2 + 4y2 = 12 4x2 + 3y2 = 1 d2 + (2b – 3c)2 = 0 d2 + (3b + 2c)2 = 0 d2(2b + 3c)2 = 0
x 2 y2 + 2 =1 2 b is Area of the greatest rectangle that can be inscribed in the ellipse a
(a) ab (b) 2ab (c)
a b
(d)
ab
[Ans. : c] 4. Let P be the point (1, 0) and Q a point on the locus y2 = 8x. The locus of mid point of PQ is (a) (c) [Ans. : b] 5. The parabolas y2 = 4x and x2 = 4y divide the square region bounded by the lines x = 4, y = 4 and the coordinates axes. If S1, S2, S3 are respectively the areas of these parts numbered from top to bottom; then S1 : S2 : S3 is (a) (c) [Ans. : c] 6. An ellipse has OB as semi minor axis, F and F′ its focii and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is (a) [Ans. : b] 7. The locus of a point P (α, β) moving under the condition that the line y = αx + β is a tangent to the x 2 y2 − 2 =1 2 b hyperbola a is (a) a circle (b) an ellipse 1:2:3 1:1:1 (b) (d) 1:2:1 2:1:2 y2 + 4x + 2 = 0 x2 – 4y + 2 = 0 (b) (d) y2 – 4x + 2 = 0 x2 + 4y + 2 = 0
1 2
(b)
1 2
(c)
1 3
(d)
1 4
(c) [Ans. : c]
a hyperbola
(d)
a parabola