The length of the latus rectum of the hyperbola $25x^2 - 16y^2 = 400$ is -

The asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,with any tangent to the hyperbola form a triangle whose area is $a^2 \tan (\alpha)$. Then its eccentricity equals

The equation of the normal to the curve $x=a \cosh(t), y=b \sinh(t)$ at any point $t$ is

The eccentricity of a rectangular hyperbola is

The locus of the point of intersection of the lines $\sqrt{3}x - y - 4\sqrt{3}k = 0$ and $\sqrt{3}kx + ky - 4\sqrt{3} = 0$ for different real values of $k$ is a hyperbola $H$. If $e$ is the eccentricity of $H$,then $4e^2 =$

If a hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$, then the tangent to this hyperbola at $P$ is: