The locus of the midpoint of the system of parallel chords parallel to the line $y = 2x$ for the hyperbola $9x^2 - 4y^2 = 36$ is

If $y=x+\sqrt{2}$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{2}=1$,then the equations of its directrices are

The asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ form with any tangent to the hyperbola a triangle whose area is $a^2 \tan \lambda$ in magnitude. Then its eccentricity $e$ is:

Let $P(4,3)$ be a point on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. If the normal at $P$ intersects the $X$-axis at $(16,0)$, then the eccentricity of the hyperbola is

Find the equation of the hyperbola whose foci are $(-2, 0)$ and $(2, 0)$ and eccentricity is $2$.

If the normal drawn to the hyperbola $xy=16$ at $(8,2)$ meets the hyperbola again at a point $(\alpha, \beta)$,then $|\beta|+\frac{1}{|\alpha|}=$