Hyperbola

If $P Q$ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $O P Q$ is an equilateral triangle, $O$ bing the centre of the hyperbola, then the eccentricity $e$ of the hyperbola satisfies.

The difference of the focal distance of any point on the hyperbola is equal to its

If $e$ is the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\theta$ is the angle between the asymptotes, then $\cos \left(\frac{\theta}{2}\right)$ is equal to

Which one of the following is independent of $\alpha$ in the hyperbola $\quad(0 \lt \alpha \lt \pi / 2)$ $\frac{x^2}{\cos ^2 \alpha}-\frac{y^2}{\sin ^2 \alpha}=1$

The diameter of $16 x^2-9 y^2=144$ which is conjugate to $x=2 y$ is

If the equation $l x^2+2 m x y+n y^2=0$ represents a pair conjugate diameter of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then

All the chords of the hyperbola $3 x^2-y^2-2 x+4 y=0$, subtending a right angle at the origin pass through the fixed point

The tangent and normal to a rectangular hyperbola $x y=c^2$ at a point cuts off intercepts $a_1$ and $a_2$ on one axis and $b_1, b_2$ on the other, then $a_1 a_2+b_1 b_2$ is equal to

If the chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4 a x$. Then, the locus of the middle points of these chords is

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

Consider the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ the area of the triangle formed by the asymptotes and the tangent drawn to it at $(a, 0)$ is

The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is 6 .The equation of the hyperbola referred to its axes as axes of coordinates are