If the eccentricities of the hyperbolas $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ are $e$ and $e_1$ respectively,then $\frac{1}{e^2} + \frac{1}{e_1^2} = $

The point of contact of the line $y = x - 1$ with the hyperbola $3x^2 - 4y^2 = 12$ is

The line $2x + y = 1$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > b)$. If this line passes through the point of intersection of a directrix and the positive $X$-axis,then the eccentricity of that hyperbola is

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with centre $C$ are such that $CP$ is perpendicular to $CQ$,where $a < b$,then the value of $\frac{1}{(CP)^2} + \frac{1}{(CQ)^2}$ is:

If the angle between the asymptotes of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \tan^{-1}\left(\frac{b}{a}\right) = 2 \tan^{-1}\left(\frac{2}{3}\right)$ and $a^2-b^2=45$,then $ab=$

The angle between the tangents drawn from $(1, 2\sqrt{2})$ to the hyperbola $16x^{2} - 25y^{2} = 400$ is.....