For a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,if the length of the transverse axis is $8$ and the distance between the foci is $2\sqrt{41}$,then the length of its latus rectum is

At which point on the curve $3x^2 - y^2 = 8$ is the normal parallel to the line $x + 3y = 4$?

The tangent to the hyperbola $xy = c^2$ at the point $P(ct, c/t)$ intersects the $x$-axis at $T$ and the $y$-axis at $T'$. The normal to the hyperbola at $P$ intersects the $x$-axis at $N$ and the $y$-axis at $N'$. If the areas of the triangles $PNT$ and $PN'T'$ are $\Delta$ and $\Delta'$ respectively,then $\frac{1}{\Delta} + \frac{1}{\Delta'}$ is:

$P(6, 3)$ is a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If the normal at point $P$ intersects the $x$-axis at $(10, 0)$,then the eccentricity of the hyperbola 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 length of the latus rectum of the hyperbola $25x^2 - 16y^2 = 400$ is -