The number of real tangents that can be drawn to the ellipse $3x^2 + 5y^2 = 32$ passing through $(3, 5)$ is

The extremities of the latera recta of the ellipses $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ having a given major axis $2a$ lie on:

The distance between the directrices of the ellipse $\frac{x^2}{36}+\frac{y^2}{20}=1$ is

An ellipse passing through $(4 \sqrt{2}, 2 \sqrt{6})$ has foci at $(-4, 0)$ and $(4, 0)$. Then,its eccentricity is

If the tangents drawn from a point $P$ to the ellipse $4 x^2+9 y^2-24 x+36 y=0$ are perpendicular,then the locus of $P$ is

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 = $