Practice Test – 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 $
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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22. The eccentricity of the conjugate hyperbola of the hyperbola x2 – 3y2 = 1 is
(A) 2
(B) 2/√3
(C) 1
(D) 4/3
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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
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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
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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
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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
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27. The curve represented by x = aeθ , y = be-θ ,θ ∈ R is
(A) a hyperbola
(B) an ellipse
(C) a parabola
(D) a circle
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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
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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
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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
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11. A 12. C 13. A 14. B 15. B 16. D 17. B 18. B 19. D 20. A
21. D 22. B 23. C 24. A 25. B 26. C 27. A 28. A 29. B 30. D