## Hyperbola Questions and Answers Part-1

1. The equation of the hyperbola whose conjugate axis is 5 and distance between the foci is 13, is
a) $25x^{2}-144y^{2}=900$
b) $144x^{2}-25y^{2}=900$
c) $25x^{2}-36y^{2}=900$
d) $36x^{2}-25y^{2}=900$

Explanation:

2. The vertices of a hyperbola are (0,0) and (10,0). If one of its foci is (18,0) , the equation of the hyperbola is
a) $\frac{x^{2}}{25}-\frac{y^{2}}{144}=1$
b) $\frac{\left(x-5\right)^{2}}{25}-\frac{y^{2}}{144}=1$
c) $\frac{x^{2}}{25}-\frac{\left(y-5\right)^{2}}{144}=1$
d) $\frac{\left(x-5\right)^{2}}{25}-\frac{\left(y-5\right)^{2}}{144}=1$

Explanation: Centre of the hyperbola is the mid point (5,0)

3.The equation of the hyperbola whose foci are (6,4) and (-4,4) and eccentricity 2 is given by
a) $12x^{2}-4y^{2}+24x-32y-127=0$
b) $12x^{2}-4y^{2}-24x+32y-127=0$
c) $12x^{2}-4y^{2}+24x+32y+127=0$
d) $12x^{2}-4y^{2}-24x+32y+127=0$

Explanation: Center is the midpoint (1,4) of the foci

4. If $m_{1},m_{2}$  are the slopes of the tangents to the hyperbola $\frac{x^{2}}{144}-\frac{y^{2}}{25}=1$     which pass through the point (14,6) , then
a) $m_{1}+m_{2}=\frac{42}{13}$
b) $m_{1}m_{2}=\frac{11}{52}$
c) $m_{1}+m_{2}=\frac{51}{52}$
d) $m_{1}m_{2}=\frac{42}{13}$

Explanation: Equation of a line through (14,6) is

5. The value of m, for which the line $y=mx+\frac{25\sqrt{3}}{3}$    is a normal to the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$
is
a) $\sqrt{3}$
b) $-\frac{2}{\sqrt{3}}$
c) $-\frac{\sqrt{3}}{2}$
d) 1

Explanation: Equation of a normal to the given hyperbola is

6. Let $P\left(a\sec \theta,b\tan\theta\right)$    and $Q\left(a\sec \phi,b\tan\phi\right)$    where $\theta+\phi=\pi/2,$    be two points on the hyperbola $x^{2}/a^{2}-y^{2}/b^{2}=1.$     If (h, k) is the point of intersection of normals at P and Q, then k is equal to
a) $\frac{a^{2}+b^{2}}{a}$
b) $-\left[\frac{a^{2}+b^{2}}{a}\right]$
c) $\frac{a^{2}+b^{2}}{b}$
d) $-\left[\frac{a^{2}+b^{2}}{b}\right]$

Explanation: Equation of the normal at P is

7. If P is a point on the rectangular hyperbola $x^{2}-y^{2}=a^{2},C$     is its centre and S, S' are the two foci, then SP. S'P =
a) 2
b) $\left(CP\right)^{2}$
c) $\left(CS\right)^{2}$
d) $\left(SS'\right)^{2}$

Explanation: Let the coordinates of P be (x, y)

8. If PQ is a double ordinate of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$    such that OPQ is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola, satisfies
a) $1< e< 2/\sqrt{3}$
b) $e=2/\sqrt{3}$
c) $e=\sqrt{3}/2$
d) $e>2/\sqrt{3}$

Explanation: Let the coordinates of P be ( $\alpha,\beta$ )

9. Let a and b be non-zero real numbers. Then the equation $\left(ax^{2}+by^{2}+c\right)\left(x^{2}-5xy+6y^{2}\right)=0$
represents
a) Four straight lines ,when c =0 and a,b are of the same sign.
b) Two straight lines and a circle ,when a=b and c and a are of the same sign.
c) Two straight lines and a hyperbola ,when a and b are of opposite sign
d) a circle and an ellipse , when a and b are of the same sign and c is of sign opposite to that of a.

10. If x= 9 is the chord of contact of the hyperbola $x^{2}-y^{2}=9$   , then the equation of the corresponding pair of tangents is
a) $9x^{2}-8y^{2}+18x-9=0$
b) $9x^{2}-8y^{2}-18x+9=0$
c) $9x^{2}-8y^{2}-18x-9=0$
d) $9x^{2}-8y^{2}+18x+9=0$