Product of length of the perpendiculars drawn from foci on any tangent to hyperbola ${x^2} - \frac{{{y^2}}}{4}$ = $1$ is

Locus of mid points of chords of hyperbola $x^2 -y^2 = a^2$ which are tangents to the parabola $x^2 = 4by$ will be -

The equation of a hyperbola, whose foci are $(5, 0)$ and $(-5, 0)$ and the length of whose conjugate axis is $8$, is

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be reciprocal to that of the ellips $x^2+4 y^2=4$. If the hyperbola passes through a focus of the ellipse, then

$(A)$ the equation of the hyperbola is $\frac{x^2}{3}-\frac{y^2}{2}=1$

$(B)$ a focus of the hyperbola is $(2,0)$

$(C)$ the eccentricity of the hyperbola is $\sqrt{\frac{5}{3}}$

$(D)$ the equation of the hyperbola is $x^2-3 y^2=3$

An ellipse intersects the hyperbola $2 x^2-2 y^2=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

$(A)$ Equation of ellipse is $x^2+2 y^2=2$

$(B)$ The foci of ellipse are $( \pm 1,0)$

$(C)$ Equation of ellipse is $x^2+2 y^2=4$

$(D)$ The foci of ellipse are $( \pm \sqrt{2}, 0)$

Find the equation of the hyperbola satisfying the give conditions: Foci $(0,\,\pm 13),$ the conjugate axis is of length $24.$