Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 3 \sqrt{5}, 0)$,the latus rectum is of length $8$.

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 the latus rectum of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends an angle of $60^{\circ}$ at the other focus,then the eccentricity of the hyperbola is

Consider a hyperbola $H$ having its centre at the origin and foci on the $x$-axis. Let $C_1$ be a circle touching the hyperbola $H$ and having its centre at the origin. Let $C_2$ be a circle touching the hyperbola $H$ at its vertex and having its centre at one of its foci. If the areas (in sq. units) of $C_1$ and $C_2$ are $36 \pi$ and $4 \pi$,respectively,then the length (in units) of the latus rectum of $H$ is

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:

If the perimeter of a triangle is $20$ and two of its vertices are $(-5, 0)$ and $(6, 0)$,then the locus of the third vertex is: