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

The equation of the tangent to the hyperbola $4{y^2} = {x^2} - 1$ at the point $(1, 0)$ is

Let the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ pass through the point $(2 \sqrt{2},-2 \sqrt{2})$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is e times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?

The locus of the point of instruction of the lines $\sqrt 3 x - y - 4 \sqrt 3 t = 0$  $\&$  $\sqrt 3tx + ty - 4\sqrt 3 = 0$  (where $ t$  is a parameter) is a hyperbola whose eccentricity is

A normal to the hyperbola, $4x^2 - 9y^2\, = 36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$, respectively . If the parallelogram $OABP$ ( $O$ being the origin) is formed, then the locus of $P$ is

The locus of a point $P (h, k)$ such that the line $y = hx + k$ is tangent to $4x^2 - 3y^2 = 1$ , is a/an