The auxiliary circle of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is given by:

Let tangents drawn from point $C(0,-b)$ to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ touch the hyperbola at points $A$ and $B$. If $\Delta ABC$ is a right-angled triangle,then $\frac{a^2}{b^2}$ is equal to -

The locus of a point which moves such that the difference of its distances from two fixed points is always a constant is

$A$ hyperbola has its transverse axis along the major axis of the conic $\frac{x^2}{3} + \frac{y^2}{4} = 4$ and its vertices at the foci of this conic. If the eccentricity of the hyperbola is $\frac{3}{2}$,then which of the following points does $NOT$ lie on it?

Let $P(3 \sec \theta, 2 \tan \theta)$ and $Q(3 \sec \phi, 2 \tan \phi)$ be two points on the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ such that $\theta + \phi = \frac{\pi}{2}$ where $0 < \theta, \phi < \frac{\pi}{2}$. Then the ordinate of the point of intersection of the normals at $P$ and $Q$ is:

Let $X$-axis be the transverse axis and $Y$-axis be the conjugate axis of a hyperbola $H$. Let the eccentricity of $H$ be the reciprocal of the eccentricity of the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$. If $(5, 4)$ is a point on $H$,then the length of the transverse axis of $H$ is