The equation of the hyperbola,whose eccentricity is $\sqrt{2}$ and whose foci are $16$ units apart,is

The equations of the tangents to the hyperbola $4x^2 - y^2 = 12$ are $y = 4x + c_1$ and $y = 4x + c_2$. Then $|c_1 - c_2|$ is equal to -

The vertices of a hyperbola $H$ are $(\pm 6, 0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $N$ be the normal to $H$ at a point in the first quadrant and parallel to the line $\sqrt{2} x + y = 2 \sqrt{2}$. If $d$ is the length of the line segment of $N$ between $H$ and the $y$-axis,then $d^2$ is equal to $............$.

$A$ hyperbola, having the transverse axis of length $2 \sin \theta$, is confocal with the ellipse $3 x^{2}+4 y^{2}=12$. Its equation is

When is the line $ℓx + my + n = 0$ a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$?

If the coordinates of a point are given by the equations $x = b \sec \phi$ and $y = a \tan \phi$,then its locus is