Untitled Document

Question Bank No: 1

1. The distance between the foci of a hyperbola is 16 and its eccentricity is $\sqrt{2}$ then the equation of hyperbola is

a)		$x^{2} + y^{2} = 32$
b)		$x^{2} - y^{2} = 16$
c)		$x^{2} + y^{2} = 16$
d)		$x^{2} - y^{2} = 32$

2. The hyperbola is the conic with eccentricity

a)		e>1
b)		e<1
c)		e=1
d)		e=0

3. The latus rectum of the hyperbola 9 $x^{2} - 16 y^{2} - 18 x - 32 y - 151 = 0 is$

a)		$\frac{9}{4}$
b)		9
c)		$\frac{3}{2}$
d)		$\frac{9}{2}$

4. The equation of the conic with focus at (1,-1) directrix along x-y+1=0 and with eccebtricity $\sqrt{2}$ is

a)		$x^{2} - y^{2} = 1$
b)		xy=1
c)		2xy-4x+4y+1=0
d)		2xy+4x-4y-1=0

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

a)		$\frac{{2 a}^{2}}{b}$
b)		$\frac{{2 b}^{2}}{a}$
c)		$\frac{b^{2}}{a}$
d)		$\frac{a^{2}}{b}$

6. If e and $e_{1}$ are the essentricities of the hyperbola xy= $c^{2}$ and $x^{2} - y^{2} = c^{2}$ then $e^{2} + {e_{1}}^{2}$ is

a)		1
b)		4
c)		6
d)		8

7. If e, $e^{'}$ are the eccentricity of hyperbolas $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 and \frac{x^{2}}{b^{2}} - \frac{y^{2}}{a^{2}} = 1$ then

a)		e = e'
b)		e= -e'
c)		e e'=1
d)		$\frac{1}{e^{2}} + \frac{1}{e^{' 2}}$ =1

8. The equation x= $\frac{e^{t} + e^{- t}}{2}$ y = $\frac{e^{t} - e^{- t}}{2}$ , t $\in R$ represents

a)		an ellipse
b)		a parabola
c)		a hyperbola
d)		a circle

9. The essentricity of the conic $x^{2}$ -2x-4 $y^{2}$ =0 is

a)		$\frac{1}{4}$
b)		$\frac{\sqrt{5}}{2}$
c)		$\frac{3}{2}$
d)		$\frac{\sqrt{5}}{4}$

10. The eccentricity of the conic 9 $x^{2}$ -16 $y^{2}$ =144 is

a)		$\frac{4}{5}$
b)		$\frac{5}{4}$
c)		$\frac{4}{3}$
d)		$\sqrt{7}$

11. For the hyperbola $\frac{x^{2}}{\cos^{2} α} - \frac{y^{2}}{\sin^{2} α} = 1,$ which of the following remains constant when $α$ varies?

a)		directrix
b)		abscissae of vertices
c)		abscissae of foci
d)		eccentricities

12. A hyperbola, having the transverse axis of length 2 sin $θ,$ is confocal with the ellipse $3 x^{2} + 4 y^{2} = 12.$ Then its equation is

a)		$x^{2} {cosec}^{2} θ - y^{2} \sec^{2} θ = 1$
b)		$x^{2} \sec^{2} θ - y^{2} {cosec}^{2} θ = 1$
c)		$x^{2} \sin^{2} θ - y^{2} \cos^{2} θ = 1$
d)		$x^{2} \cos^{2} θ - y^{2} \sin^{2} θ = 1$

13. The locus of a point $P (α, β)$ moving under the condition that the line y= $α x + β$ is a tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is

a)		circle
b)		ellipse
c)		parabola
d)		hyperbola

14. If a hyperbola passes through a focus of the ellipse and its transverse and conjugate axes coincide with major and minor axis of the ellipse and the product of their eccentricities is 1, then

a)		the equation of the hyberbola is $\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$
b)		the equation of the hyberbola is $\frac{x^{2}}{9} - \frac{y^{2}}{25} = 1$
c)		focus of hyberbola is (-5,0)
d)		focus of hyberbola is (5 $\sqrt{3}$ ,0)

15. Tangents are drawn from points on the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$ to tje circle $x^{2} + y^{2} = 9.$ The locus of the midpoint of the chord of contact is

a)		$x^{2} + y^{2} = \frac{x^{2}}{9} - \frac{y^{2}}{4}$
b)		${(x^{2} + y^{2})}^{2} = \frac{x^{2}}{9} - \frac{y^{2}}{4}$
c)		${(x^{2} + y^{2})}^{2} = 81 (\frac{x^{2}}{9} - \frac{y^{2}}{4})$
d)		${(x^{2} + y^{2})}^{2} = 9 (\frac{x^{2}}{9} - \frac{y^{2}}{4})$

16. If the pair of lines $b^{2} x^{2} - a^{2} y^{2} = 0$ are inclined at an angle $θ,$ then teh eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is

a)		sec $θ$
b)		cos $θ$
c)		cos $\frac{θ}{2}$
d)		sec $\frac{θ}{2}$

17. The hyberbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ has its conjugate axis 5 and passes through the point (2, -1). The length of a latus rectum is

a)		$\frac{5}{4} \sqrt{29}$
b)		$\frac{5}{8} \sqrt{29}$
c)		$\sqrt{29}$
d)		$\frac{\sqrt{29}}{4}$

18. The distance between foci of $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 is 9$ and eccentricity is $\sqrt{3.}$ The length of a latus-rectum is

a)		$3 \sqrt{3}$
b)		$6 \sqrt{3}$
c)		3
d)		6

19. If x=9 is the chord of contact of tangents of $x^{2} - y^{2} = 9,$ then the equation of the corresponding tangents is

a)		$9 x^{2} - 8 y^{2} + 18 x + 9 = 0$
b)		$9 x^{2} - 8 y^{2} - 18 x + 9 = 0$
c)		$9 x^{2} - 8 y^{2} - 18 x - 9 = 0$
d)		$9 x^{2} - 8 y^{2} + 18 x - 9 = 0$

20. Let P (a sec $θ, b \tan θ)$ and Q=(a sec $φ, b \tan φ)$ where $θ + φ = \frac{π}{2}$ be two points on $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.$ If (h, k) is the point of intersection of normals at P and Q, then k is

a)		$\frac{{a^{2} + b}^{2}}{a}$
b)		$- \frac{({a^{2} + b}^{2})}{a}$
c)		$\frac{{a^{2} + b}^{2}}{b}$
d)		$- \frac{({a^{2} + b}^{2})}{b}$

21. The foci of the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{b^{2}} = 1$ and the hyperbola $\frac{x^{2}}{144} - \frac{y^{2}}{81} = \frac{1}{25}$ coincide. Then the value of $b^{2} is$

a)		1
b)		5
c)		7
d)		9

22. If the line $2 x + \sqrt{6} y = 2$ touches the hyperbola $x^{2} - 2 y^{2} = 4,$ then the point of contact is

a)		$(- 2, \sqrt{6})$
b)		$(- 5, 2 \sqrt{6})$
c)		$(\frac{1}{2}, \frac{1}{\sqrt{6}})$
d)		$(4, - \sqrt{6})$

23. Given the equation of the hyberbola $\frac{x^{2}}{\cos^{2} α} - \frac{y^{2}}{\sin^{2} α} = 1,$ $o < α < \frac{π}{2}$ , which of the following does not depend on $α ?$

a)		foci
b)		eccentricity
c)		directrices
d)		vertices

24. If the hyperbola xy= $c^{2}$ intersects the circle $x^{2} + y^{2} = a^{2}$ in four points $P (x_{1}, y_{1}), Q (x_{2}, y_{2})$ , $R (x_{3}, y_{3})$ and $S (x_{4}, y_{4})$ , then

a)		$x_{1} + x_{2} + x_{3} = 0$
b)		$y_{1} + y_{2} + y_{3} = 0$
c)		$x_{1} x_{2} x_{3} = c^{4}$
d)		$y_{1} y_{2} y_{3} y_{4} = c^{4}$

25. If the line ax + by + c = 0 is a normal to the hyperbola xy=1, then

a)		$a > 0, b > 0$
b)		0 < a < 1
c)		$a < 0, b > 0$
d)		$a < 0, b < 0$