Single Answer Type Questions
1. If the tangent at on the ellipse cuts auxiliary circle at points A and B. If C is the centre of the ellipse, then the area of the triangle CAB (in square units) is
2. A hyperbola passes through the points (3, 2) and (-17, 12) and has its centre at the origin and transverse axis along X-axis, then the length of its transverse axis is .......
3. A variable straight line of slope 4 intersects the hyperbola xy = 1 at two points, the locus of the point which divides the line segment between these points in the ratio 1 : 2 is
Ans: 16x2 + y2 + 10xy = 2
4. If the normals at the points Pi (xi, yi), i = 1 to 4 on the hyperbola xy = are concurrent at the point Q(h, k), then equal to
5. Angle between the asymptotes of hyperbola x2 + 2xy - 3y2 + x + 7y + 9 = 0 is
Ans: −tan−1 2
6. The area of triangle formed by the lines x − y = 0, x + y = 0 and any tangent to the hyperbola x2 − y2 = a2 is
7. Eccentricity of the hyperbola conjugate to the hyperbola is
8. A hyperbola having the transverse axis of length 2 sin θ is confocal with the ellipse 3x2 + 4y2 = 12 then its equation is
Ans: x2cosec2 θ − y2sec2 θ = 1
9. The locus of the middle points of chords of hyperbola 3x2 - 2y2 + 4x − 6y = 0 parallel to y = 2x is
Ans: 3x - 4y = 4
10. If α, β are eccentric angles of the focal chord of the hyperbola then tan tan =
11. A hyperbola has eccentricity 'e' satisfying e2 − ( + 1)e + = 0. The eccentricity of its conjugate hyperbola is ....
12. Consider a branch of the hyperbola x2 - 2y2 - 2 x - 4 y − 6 = 0 with the vertex at the point A, let B be one end of latus rectum, if C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is
13. The shortest distance between the curves , 4x2 + 4y2 = a2 (b > a) is ......
14. Area of the triangle formed by any arbitary tangent of the hyperbola xy = c2 with coordinate axes, is equal to ......
15. The normal at 't' to xy = c2 meets it again at t1 then t3t1 =
16. The point (3 tan (θ + 60°), 2 tan (θ + 30°)) lies on the hyperbola, then its centre is (θ is parameter)
Ans: (−3 , 2 )
17. In the hyperbola xy = 4 and the locus of middle points of chords of contact of length 2 is ....
Ans: (x2 + y2)(xy - 4) = xy
18. Consider a hyperbola xy = 4 and a line y + 2x = 4. Let the given line intersect X - axis at R. If a line through R intersect hyperbola at S and T, the minimum value of RS. RT is ......
19. A rectangular hyperbola whose centre is C cut by any circle of radius r in four points P, Q, R, S then CP2 + CQ2 + CR2 + CS2 equals to ....
20. Let P(a sec θ, b tan θ) and P(a sec ∅, b tan ∅ ) where θ + ∅ = be two points on the hyperbola . If (h, k) is the point of intersection of normals at P and Q, then k =
Multiple Correct Answer Type Questions
21. The equation (x − α)2 + (y − β)2 = k(lx + my + n)2 represents
A) A parabola for k < (l2 + m2)−1 B) An ellipse for < k < (l2 + m2)−1
C) A hyperbola for k > (l2 + m2)−1 D) A point circle for k = 0
Ans: B, C, D
22. The equation of the tangent to the hyperbola 3x2 − 4y2 = 12 which make equal intercepts on the axes is
A) y = x + 1 B) y = x − 1 C) y = −x + 1 D) y = −x −1
Ans: A, B, C, D
23. A foci of the hyperbola 25x2 − 36y2 = 225 is ....
A) (, 0) B) C) (−, 0) D)
Ans: B, D
24. If the normal at P to the rectangular hyperbola x2 − y2 = 4 meets the axes in G and g and C be the centre of the hyperbola, then
A) PG = PC B) Pg = PC C) PG = Pg D) Gg = 2PC
Ans: A, B, C, D
25. For the rectangular hyperbola xy = c2, which of the following is true?
A) Equation of chord joining (x1, y1) and (x2, y2) is
B) Equation of chord joining (x1, y1), (x2, y2) is
C) Tangents from origin are perpendicular
D) Tangents from origin are inclined at 60°
Ans: A, C
26. If (a > b) and a2 − y2 = c2 cut at right angles, then
A) a2 + b2 = 2c2 B) b2 − a2 = 2c2 C) a2 − b2 = 2c2 D) a2 > 2c2
Ans: C, D
27. The curves x = t2 + 1, y = 2t and x = 2s, y = where (t, s are parameters)
A) meet at only one point B) meet at two points
C) meet at (2, 2) D) meet at (1, 2)
Ans: A, C
28. An ellipse intersects the hyperbola 2x2 − 2y2 = 1 orthogonally. The eccentricity of the ellipse is reciprocal of that the hyperbola, if the axes of the ellipse are along the coordinate axes, thena
A) Equation of ellipse is x2 + 2y2 = 2 B) The foci of ellipse are (± 1, 0)
C) Equation of ellipse is x2 + 2y2 = 4 D) The foci of ellipse are (± , 0)
Ans: A, B
29. The lines parallel to normal to the curve xy = 1, is/are
A) 3x + 4y + 5 =0 B) 3x − 4y + 5 = 0
C) 4x + 3y + 5 =0 D) 3y − 4x + 5 = 0
Ans: B, D
30. If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2), R(x3, y3), S(x4, y4), then
A) x1 + x2 + x3 + x4 = 0 B) y1 + y2 + y3 + y4 = 0
C) x1 x2 x3 x4 = c4 D) y1 y2 y3 y4 = c4
Ans: A, B, C, D
Passage - I
If the normal to the hyperbola at a point A (a sec θ, b tan θ) meets the transverse and conjugate axes in D and E respectively and F is the foot of the perpendicular to the normal at A from the centre C, then
31. The value of AD2
32. Minimum value of AD is
A) B) C) D)
33. The value of AE2 is
Passage - II
A line drawn through P(−1, 2) meets the hyperbola xy = c2 at the points A and B (Points A & B lie on the same side of P)
34. A point 'Q' is chosen on this line such that PA, PQ, PB are in A.P., then locus of point Q is ....
A) x = y(1 + 2x) B) x = y(1 + x) C) 2x = y(1 + 2x) D) x = 2y(1 + 2x)
35. If PA, PQ, PB are in G.P., then locus of point Q is
A) xy − y + 2x − c2 =0 B) xy + y − 2x + c2 = 0
C) xy + y + 2x + c2 =0 D) xy − y − 2x − c2 = 0
36. If PA, PQ and PB are in H.P., then locus of point Q is ....
A) 2x − y = 2c2 B) x − 2y = 2c2 C) 2x + y = 2c2 D) x + 2y = 2c2
Matrix Match Type Questions
Let the circle (x − 1)2 + (y − 2)2 = 25 cut a rectangular hyperbola with transverse axis along y = x at four points A, B, C and D having co-ordinates (xi, yi); i = 1, 2, 3, 4 respectively origin 'O' being the centre of the four hyperbola. Now match the entries from the following two columns.
Ans: A→p, B→ r, C→q, D→ s
Ans: A → s, B→ q, C→ r , D → p
Integer Answer Type questions
39. If the tangent and normal to the rectangular hyperbola at a point cut off intercepts a1, a2 on X−axis and b1, b2 on the Y − axis, then a1a2 + b1b2 = ......
40. PM and PN are the perpendiculars from any point on the rectangular hyperbola xy = c2 to the asymptotes. The locus of the midpoint of MN is a hyperbola with eccentricity k, then k2 = ........