## Hyperbola Questions and Answers Part-3

1. If the angle between a pair of tangents from a point P to the parabola $y^{2}=4ax$   is $\frac{\pi}{4}$   then the locus of P is a hyperbola whose equation is
a) $x^{2}-y^{2}=8a^{2}$
b) $\left(x-3a\right)^{2}-y^{2}=8a^{2}$
c) $\left(x+3a\right)^{2}-y^{2}=8a^{2}$
d) $x^{2}-y^{2}=a^{2}$

Explanation: Let P( $\alpha,\beta$ ) be any point on the locus. Equation of pair of tangents from P( $\alpha,\beta$ ) to the parabola y2 = 4ax is

2. Tangents are drawn to the circle $x^{2}+y^{2}=9$   from points on the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1.$    Locus of the mid point of the chords of contact is
a) a circle
b) an ellipse
c) a hyperbola
d) none of these

Explanation:

3. If $0<\alpha<\pi/2,\alpha\neq\frac{\pi}{4}$     then for the hyperbola $\frac{x^{2}}{\left(\cos\alpha+\sin\alpha\right)^{2}}-\frac{y^{2}}{\left(\cos\alpha-\sin\alpha\right)^{2}}=1$
which one of the following is independent of $\alpha$
a) eccentricity
b) directrix
c) abscissa of vertices
d) acissa of foci

Explanation:

4. All chords of the curve $3x^{2}-y^{2}-2x+4y=0$     which subtend a right at the origin pass through
a) centre of the rectangular hyperbola $x^{2}-y^{2}-2x-4y=12$
b) the point of intersection of the lines y+2x=0 and x=1
c) the vertex of the parabola $x^{2}-2x-4y-7=0$
d) All of the Above

Explanation: Let y = mx + c be a chord of the given curve.

5. The equation of a tangent to the hyperbola $3x^{2}-y^{2}=3$   , parallel to the line y = 2x + 4 is
a) y = 2x + 3
b) y = 2x + 1
c) y = 2x - 1
d) Both b and c

Explanation: The equation of the hyperbola is

6. If a variable straight line x cos $\alpha+y\sin \alpha=p$    which is a chord of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$     ( b > 0) subtend a right angle at the centre of the hyperbola , then it always touches a fixed circle whose
a) centre is the centre of the hyperbola
b) radius is $\frac{ab}{\sqrt{b^{2}-a^{2}}}$
c) centre (0,0)
d) All of the Above

Explanation: Equation of the pair of straight lines passing

7. An ellipse intersects the hyperbola $2x^{2}-2y^{2}=1$    orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then
a) Equation of ellipse is $x^{2}+2y^{2}=2$
b) The foci of ellipse are $\left(\pm1,0\right)$
c) Equation of ellipse is $x^{2}+2y^{2}=4$
d) Both a and b

Explanation:

8. If the circle $x^{2}+y^{2}=a^{2}$   intersect the hyperbola $xy=c^{2}$   in four points $P\left(x_{1},y_{1}\right),Q\left(x_{2},y_{2}\right),R\left(x_{3},y_{3}\right),S\left(x_{4},y_{4}\right)$
then
a) $x_{1}+x_{2}+x_{3}+x_{4}=0$
b) $x_{1}x_{2}x_{3}x_{4}=e^{4}$
c) $y_{1}+y_{2}+y_{3}+y_{4}=0$
d) All of the Above

Explanation: The abscissa of the points of intersection of

9. If the normals at four points $\left(x_{1},y_{1}\right),\left(x_{2},y_{2}\right),\left(x_{3},y_{3}\right), and \left(x_{4},y_{4}\right)$
on the rectangular hyperbola xy = c2 meets at point (h, k) , then
a) $x_{1}+x_{2}+x_{3}+x_{4}=h$
b) $y_{1}+y_{2}+y_{3}+y_{4}=k$
c) $x_{1}x_{2}x_{3}x_{4}=-c^{4}$
d) All of the Above

10. Equation of a tangent passing throught (2,8) to the hyperbola $5x^{2}-y^{2}=5$
a) $3x-y+2=0$
b) $3x+y+14=0$
c) $23x-3y-22=0$