If $2x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{16} = 1$,then which of the following $CANNOT$ be sides of a right-angled triangle?

Let $e$ be the eccentricity of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$. If $\frac{1}{e}$ is the eccentricity of a hyperbola,then the eccentricity of its conjugate hyperbola is

If the length of the latus rectum of a hyperbola is $8$ and its eccentricity is $\frac{3}{\sqrt{5}}$,then the equation of the hyperbola is:

If $P(\frac{\pi}{4})$ and $Q(\frac{3 \pi}{4})$ are two points on the hyperbola $4 x^2-y^2-8 x-2 y-13=0$ in parametric form,then the distance between $P$ and $Q$ is

The length of the latus rectum and directrices of a hyperbola with eccentricity $e$ are $9$ and $x = \pm \frac{4}{\sqrt{13}}$,respectively. Let the line $y - \sqrt{3}x + \sqrt{3} = 0$ touch this hyperbola at $(x_0, y_0)$. If $m$ is the product of the focal distances of the point $(x_0, y_0)$,then $4e^2 + m$ is equal to ...........

If a directrix of a hyperbola centered at the origin and passing through the point $(4, -2\sqrt{3})$ is $5x = 4\sqrt{5}$ and its eccentricity is $e$,then