If the lengths of the transverse and conjugate axes of a hyperbola are $8$ and $6$ respectively,find the difference of the distances of any point on the hyperbola from its foci.

If the latus rectum subtends a right angle at the center of the hyperbola,then its eccentricity is

The distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$. Its equation is

The distance between the tangents to the hyperbola $\frac{x^2}{20} - \frac{3y^2}{4} = 1$ which are parallel to the line $x + 3y = 7$ is

For different values of $\alpha$, the locus of the point of intersection of the two straight lines $\sqrt{3} x - y - 4 \sqrt{3} \alpha = 0$ and $\sqrt{3} \alpha x + \alpha y - 4 \sqrt{3} = 0$ is

Let $LL^{\prime}$ be the latus rectum through the focus $S$ of a hyperbola and $A^{\prime}$ be the opposite vertex of the hyperbola. If triangle $A^{\prime}LL^{\prime}$ is equilateral,then the eccentricity of the hyperbola is