If the line $x \cos \alpha + y \sin \alpha = 2 \sqrt{3}$ is a tangent to the ellipse $\frac{x^2}{16} + \frac{y^2}{8} = 1$ and $\alpha$ is an acute angle,then $\alpha = $

For which of the following curves is the line $x+\sqrt{3} y=2 \sqrt{3}$ a tangent at the point $\left(\frac{3 \sqrt{3}}{2}, \frac{1}{2}\right)$?

If $S$ and $S^{\prime}$ are the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and if $PSP^{\prime}$ is a focal chord with $SP=8$,then $SS^{\prime}$ is equal to

Consider the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let $S(p, q)$ be a point in the first quadrant such that $\frac{p^2}{9}+\frac{q^2}{4}>1$. Two tangents are drawn from $S$ to the ellipse,of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $T$ in the fourth quadrant. Let $R$ be the vertex of the ellipse with positive $x$-coordinate and $O$ be the center of the ellipse. If the area of the triangle $\triangle ORT$ is $\frac{3}{2}$,then which of the following options is correct?

Find the equation for the ellipse that satisfies the given conditions: Length of major axis $= 26$,foci $= (\pm 5, 0)$.

If an ellipse with its axes as coordinate axes,$2a$ and $2b$ as the lengths of its major and minor axes respectively,passes through the points $(2,2)$ and $(3,1)$,then $3a^2+5b^2=$