Find the number of points on the ellipse $\frac{x^{2}}{50} + \frac{y^{2}}{20} = 1$ from which a pair of perpendicular tangents can be drawn to the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$.

Let $E$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P, Q$ and $Q^{\prime}$ on $E$,let $M(P, Q)$ be the mid-point of the line segment joining $P$ and $Q$,and $M(P, Q^{\prime})$ be the mid-point of the line segment joining $P$ and $Q^{\prime}$. Then the maximum possible value of the distance between $M(P, Q)$ and $M(P, Q^{\prime})$,as $P, Q$ and $Q^{\prime}$ vary on $E$,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

$A$ line passing through the endpoint of the major axis $A$ and the endpoint of the minor axis $B$ of an ellipse $\frac{x^2}{9} + y^2 = 1$ touches its auxiliary circle at point $M$. Find the area of the triangle with vertices $A, M,$ and the origin $O$.

Let the ellipse $E : x^2 + 9y^2 = 9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A, P$ and the origin $O$ is $\frac{m}{n}$,where $m$ and $n$ are coprime,then $m - n$ is equal to

The equations of the directrices of the ellipse $16x^2 + 25y^2 = 400$ are