The eccentricity of the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{{25}} = 1$ is

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 sound of a cannon firing is heard one second later at a position $B$ that at position $A$. If the speed of sound is uniform, then

The equation of the common tangent to the curves $y^2 = 8x$ and $xy = -1$ is

The tangent to the hyperbola, $x^2 - 3y^2 = 3$  at the point $\left( {\sqrt 3 \,\,,\,\,0} \right)$ when associated with two asymptotes constitutes :

Eccentricity of the rectangular hyperbola $\int_0^1 {{e^x}\left( {\frac{1}{x} - \frac{1}{{{x^3}}}} \right)} \;dx$ is