**LEVEL – I **

1. The equation to the hyperbola of given transverse axis 2a along x-axis and whose vertex bisects the distance between the centre and the focus is

(A) $\frac{x^2}{a^2}-\frac{y^2}{2a^2} = 1 $

(B) $\frac{x^2}{a^2}-\frac{y^2}{3a^2} = 1 $

(C) $\frac{x^2}{a^2}-\frac{y^2}{4a^2} = 1 $

(D) $\frac{x^2}{a^2}-\frac{y^2}{a^2/4} = 1 $

2. Let (5 tan θ, 3 sec θ) be a point on the hyperbola for all values of θ ≠ (2n + 1)π/2 , then find the eccentricity of the hyperbola is

(A) 5/3

(B) √(5/3)

(C) √34/9

(D) 9/√13

3. If t is a non-zero parameter then the point $(\frac{a}{2}(t+\frac{1}{t}) , \frac{b}{2}(t-\frac{1}{t}) )$ lies on

(A) circle

(B) parabola

(C) ellipse

(D) hyperbola

4. The locus of the points of intersection of the lines √3 x – y – 4√3t and √3t x + ty – 4√3 , for different values of t is a curve of eccentricity equal to

(A) √2

(B) 2

(C) 2/√3

(D) 4√3

5. The equation of the hyperbola whose foci are (6, 5), (-4, 5) and eccentricity 5/4 is

(A) $\frac{(x-1)^2}{16}-\frac{(y-5)^2}{9} = 1 $

(B) $\frac{x^2}{16}-\frac{y^2}{9} = 1 $

(C) $\frac{(x-1)^2}{16}-\frac{(y-5)^2}{9} = -1 $

(D) None of these

6. The eccentricity of the hyperbola whose latus-rectum is 8 and conjugate axis is equal to half the distance between the foci, is

(A) 4/3

(B) 4/√3

(C) 2/√3

(D) none of these

7. Equation of the hyperbola passing through the point (1, -1) and having asymptotes x + 2y + 3 = 0 and 3x + 4y + 5 = 0 is

(A) 3x^{2} + 10xy + 8y^{2} + 14x + 22y + 7 = 0

(B) 3x^{2} – 10xy + 8y^{2} + 14x + 22y + 7 = 0

(C) 3x^{2} – 10xy + 8y^{2} – 14x + 22y + 7 = 0

(D) None of these

8. If the foci of the hyperbola $\frac{x^2}{144}-\frac{y^2}{81} = \frac{1}{25} $ coincide, with the ellipse $\frac{x^2}{16} + \frac{y^2}{b^2} = 1 $

then the value of b^{2} is

(A) 1

(B) 5

(C) 7

(D) 9

9. Let P(a secθ, b tanθ) and Q(a secφ, b tanφ) where θ + φ = π/2 , be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ . If (h, k) is points of intersection of normals at P and Q then k is equal to

(A) (a^{2}+b^{2})/a

(B) – (a^{2}+b^{2})/a

(C) (a^{2}+b^{2})/b

(D) – (a^{2}+b^{2})/b

10. The locus of the point from which the tangent can be drawn to the different branches of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ is

(A) k^{2}/b^{2} – h^{2}/a^{2} < 0

(B) k^{2}/b^{2} – h^{2}/a^{2} = 0

(C) k^{2}/b^{2} – h^{2}/a^{2} > 0

(D) none of these

__ANSWER:__

**1. B 2. C 3. D 4. B 5. A 6. C 7. A 8. C 9. B 10. C **

**LEVEL – I **

11. The equation of hyperbola whose foci are (6, 4) and (-4, 4) and eccentricity is 2 is

(A) 12(x – 1)^{2} – 4(y – 4)^{2} = 75

(B) 4(x – 1)^{2} – 12(y – 4)^{2} = 75

(C) 12(x – 4)^{2} – 4(y – 1)^{2} = 75

(D) 4(x – 4)^{2} – 12(y – 1)^{2} = 75

12. The focus of the rectangular hyperbola (x + 4) (y – 4) = 16

(A) (-4+4√2 , 4-4√2)

(B) (-4- 4√2 , 4+4√2)

(C) (-4+4√2 , 4+4√2)

(D) none of these

13. If the line y = mx + √(a^{2}m^{2} -b^{2}) touches the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ at the point (a sec θ, b tan θ) , then θ is equal to

(A) sin^{-1}(b/am)

(B) sin^{-1}(am/b)

(C) cos^{-1}(am/b)

(D) none of these

14. The line y = 4x + c touches the hyperbola x^{2} – y^{2} = 1 iff

(A) c = 0

(B) c = ±√15

(C) c = ±√2

(D) none of these

15. Consider the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $

Area of the triangle formed by the asymptotes and the tangent drawn to it at (a, 0) is

(A) ab/2

(B) ab

(C) 2ab

(D) 4ab

16. Number of point(s) outside the hyperbola $\frac{x^2}{25}-\frac{y^2}{36} = 1 $ from where two perpendicular tangents can be drawn to the hyperbola is(are)

(A) 3

(B) 2

(C) 1

(D) 0

17. If e is the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ and θ be the angle between the asymptotes, then cos(θ/2) is equal to,

(A) 1/2e

(B) 1/e

(C) 1/e^{2}

(D) none of these

18. A normal to the parabola y^{2} = 4ax with slope ‘m’ touches the rectangular hyperbola x^{2} – y^{2} = a^{2} if.

(A) m^{6} + 4m^{4} – 3m^{2} + 1 = 0

(B) m^{6} + 4m^{4} + 3m^{2} + 1 = 0

(C) m^{6} – 4m^{4} + 3m^{2} – 1 = 0

(D) m^{6} – 4m^{4} – 3m^{2} + 1 = 0

19. If the tangent and the normal to a rectangular hyperbola xy = c^{2} , at a point, cuts off intercepts a1, and a2 on the x-axis and b1, b2 on the y-axis, then a1a2 + b1 b2 is equal to

(A) 3

(B) 1

(C) 2

(D) none of these

20. The length of latus rectum of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2} = -1 $ is

(A) 2a^{2}/b

(B) 2b^{2}/a

(C) b^{2}/a

(D) a^{2}/b

__ANSWER:__

**11. A 12. C 13. A 14. B 15. B 16. D 17. B 18. B 19. D 20. A **

**LEVEL – I **

21. Locus of the points of intersection of perpendicular tangents to $\frac{x^2}{9}-\frac{y^2}{16} = 1 $ is

(A) x^{2} + y^{2} = 25

(B) x^{2} + y^{2} = 7

(C) x^{2} + y^{2} = 49

(D) none of these

22. The eccentricity of the conjugate hyperbola of the hyperbola x^{2} – 3y^{2} = 1 is

(A) 2

(B) 2/√3

(C) 1

(D) 4/3

23. The product of perpendiculars drawn from any point on a hyperbola to its asymptotes is

(A) ab/(√a+√b)

(B) ab/(a^{2}+b^{2})

(C) a^{2}b^{2}/(a^{2}+b^{2})

(D) (a^{2}+b^{2})/a^{2}b^{2}

24. A normal to the hyperbola x^{2} – 4y^{2} = 4 meets the x and y axes at A and B. The locus of the point of intersection of the straight lines drawn through A and B perpendicular to the x and y-axes respectively is

(A) 4x^{2} – y^{2} = 25

(B) 4x^{2} + y^{2} = 25

(C) x^{2}+ 4y^{2} = 25

(D) x^{2} + y^{2} = 25

25. The locus of mid-point of the portion of a line of constant slope ‘m’ between two branches of a rectangular hyperbola xy = 1 is

(A) y – mx = 0

(B) y + mx = 0

(C) my + x = 0

(D) y = x

26. The angle between the tangents drawn from any point on the circle x^{2} + y^{2} = 3 to the hyperbola x^{2}/4 – y^{2} = 1is

(A) π/3

(B) π/4

(C) π/2

(D) π/6

27. The curve represented by x = ae^{θ} , y = be^{-θ} ,θ ∈ R is

(A) a hyperbola

(B) an ellipse

(C) a parabola

(D) a circle

28. The eccentricity of the hyperbola with latus rectum 12 and semi-conjugate axis 2√3 , is

(A) 2

(B) 3

(C) √3/2

(D) 2√3

29. Let any double ordinate PNP’ of the hyperbola $\frac{x^2}{25}-\frac{y^2}{18} = 1 $ be produced both sides to meet the asymptotes in Q and Q’, then PQ . P’Q is equal to

(A) 25

(B) 18

(C) 41

(D) None of these

30. Centre of the hyperbola x^{2} + 4y^{2} + 6xy + 8x – 2y + 7 = 0 is,

(A) (1, 1)

(B) (0, 2)

(C) (2, 0)

(D) none of these

__ANSWER:__

**21. D 22. B 23. C 24. A 25. B 26. C 27. A 28. A 29. B 30. D **