Let the tangent and normal at the point $(3 \sqrt{3}, 1)$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{4}=1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $A B$ as a diameter and the line $x =2 \sqrt{5}$ intersect $C$ at the points $P$ and $Q$. If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha, \beta)$, then $\alpha^2-\beta^2$ is equal to

If the tangents on the ellipse $4x^2 + y^2 = 8$ at the points $(1, 2)$ and $(a, b)$ are perpendicular to each other, then $a^2$ is equal to

The distance between the foci of the ellipse $3{x^2} + 4{y^2} = 48$ is

Number of points on the ellipse $\frac{{{x^2}}}{{50}} + \frac{{{y^2}}}{{20}} = 1$ from which pair of perpendicular tangents are drawn to the ellips $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{9}} = 1$

If the length of the latus rectum of an ellipse is $4\,units$ and the distance between a focus and its nearest vertex on the major axis is $\frac {3}{2}\,units$ , then its eccentricity is?