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 $M \left( P , Q ^{\prime}\right)$ 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 $M \left( P , Q ^{\prime}\right)$, as $P, Q$ and $Q^{\prime}$ vary on $E$, is. . . . .

A tangent is drawn to the ellipse $\frac{{{x^2}}}{{32}} + \frac{{{y^2}}}{8} = 1$ from the point $A(8, 0)$ to touch the ellipse at point $P.$ If the normal at $P$ meets the major axis of ellipse at point $B,$ then the length $BC$ is equal to (where $C$ is centre of ellipse) - ............ $\mathrm{units}$

The length of the latus rectum of an ellipse is $\frac{1}{3}$ of the major axis. Its eccentricity is

In the ellipse, minor axis is $8$ and eccentricity is $\frac{{\sqrt 5 }}{3}$. Then major axis is

The length of the latus rectum of the ellipse $\frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{49}} = 1$

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