$A$ point moves in such a way that the difference of its distances from two points $(8,0)$ and $(-8,0)$ always remains $4$. Then, the locus of the point 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 the latus rectum subtends a right angle at the center of the hyperbola,then its eccentricity is

The locus of the point of intersection of the tangents drawn at the extremities of a normal chord of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is

$A$ 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

The equation of the chord of the hyperbola $25x^{2} - 16y^{2} = 400$ whose midpoint is $(5, 3)$ is: