MCQ : Hyperbola

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) 3x2 + 10xy + 8y2 + 14x + 22y + 7 = 0

(B) 3x2 – 10xy + 8y2 + 14x + 22y + 7 = 0

(C) 3x2 – 10xy + 8y2 – 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 b2 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) (a2+b2)/a

(B) – (a2+b2)/a

(C) (a2+b2)/b

(D) – (a2+b2)/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) k2/b2 – h2/a2 < 0

(B) k2/b2 – h2/a2 = 0

(C) k2/b2 – h2/a2 > 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 + √(a2m2 -b2) 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 x2 – y2 = 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/e2

(D) none of these

18. A normal to the parabola y2 = 4ax with slope ‘m’ touches the rectangular hyperbola x2 – y2 = a2 if.

(A) m6 + 4m4 – 3m2 + 1 = 0

(B) m6 + 4m4 + 3m2 + 1 = 0

(C) m6 – 4m4 + 3m2 – 1 = 0

(D) m6 – 4m4 – 3m2 + 1 = 0

19. If the tangent and the normal to a rectangular hyperbola xy = c2 , 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) 2a2/b

(B) 2b2/a

(C) b2/a

(D) a2/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) x2 + y2 = 25

(B) x2 + y2 = 7

(C) x2 + y2 = 49

(D) none of these

22. The eccentricity of the conjugate hyperbola of the hyperbola x2 – 3y2 = 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/(a2+b2)

(C) a2b2/(a2+b2)

(D) (a2+b2)/a2b2

24. A normal to the hyperbola x2 – 4y2 = 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) 4x2 – y2 = 25

(B) 4x2 + y2 = 25

(C) x2+ 4y2 = 25

(D) x2 + y2 = 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 x2 + y2 = 3 to the hyperbola x2/4 – y2 = 1is

(A) π/3

(B) π/4

(C) π/2

(D) π/6

27. The curve represented by x = aeθ , y = be ,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 x2 + 4y2 + 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  

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