If $P(x, y)$,$F_1 = (3, 0)$,$F_2 = (-3, 0)$ and $16x^2 + 25y^2 = 400$,then $PF_1 + PF_2 = \dots$

$A$ rod $AB$ of length $15 \ cm$ rests between two coordinate axes such that the end point $A$ lies on the $x-$axis and the end point $B$ lies on the $y-$axis. $A$ point $P(x, y)$ is taken on the rod such that $AP = 6 \ cm$. Show that the locus of $P$ is an ellipse.

If the line $x \cos \alpha + y \sin \alpha = p$ is normal to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,then

The locus of the feet of the perpendiculars from either focus of the ellipse $(x - y + 1)^2 + (2x + 2y - 6)^2 = 20$ onto any tangent is:

If the line joining the points $A(\alpha)$ and $B(\beta)$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ is a focal chord,then one possible value of $\cot \frac{\alpha}{2} \cdot \cot \frac{\beta}{2}$ is

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