The vertices of a hyperbola $H$ are $(\pm 6,0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $N$ be the normal to $H$ at a point in the first quadrant and parallel to the line $\sqrt{2} x + y =2 \sqrt{2}$. If $d$ is the length of the line segment of $N$ between $H$ and the $y$-axis then $d ^2$ is equal to $............$.

The foci of the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{{b^2}}} = 1$ and the hyperbola $\frac{{{x^2}}}{{144}} - \frac{{{y^2}}}{{81}} = \frac{1}{{25}}$ coincide. Then the value of $b^2$ is -

Let the equation of two diameters of a circle $x ^{2}+ y ^{2}$ $-2 x +2 fy +1=0$ be $2 px - y =1$ and $2 x + py =4 p$. Then the slope $m \in(0, \infty)$ of the tangent to the hyperbola $3 x^{2}-y^{2}=3$ passing through the centre of the circle is equal to $......$

If $2 x-y+1=0$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{16}=1$, then which of the following $CANNOT$ be sides of a right angled triangle?

$[A]$ $2 a, 4,1$   $[B]$ $2 a, 8,1$   $[C]$ $a, 4,1$    $[D]$ $a, 4,2$

If the latus rectum of an hyperbola be 8 and eccentricity be $3/\sqrt 5 $, then the equation of the hyperbola is

The equation of a line passing through the centre of a rectangular hyperbola is $x -y -1 = 0$. If one of the asymptotes is $3x -4y -6 = 0$, the equation of other asymptote is