If the eccentricity of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b)$ is $e = 2/3$ and its focal chord is $3x + 2y - 6 = 0$, then the value of $a^2 + b^2$ is...

Let $C$ be the centre of an ellipse and $PQ$ be a chord of it with $\angle PCQ = 90^{\circ}$. If $R$ is the point of intersection of the tangents to the ellipse at $P$ and $Q$,then $R$ lies on

Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 5, 0)$,foci $(\pm 4, 0)$.

The length of the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$,whose mid-point is $\left(1, \frac{1}{2}\right)$,is:

If $P$ lies in the first quadrant on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b$),and the tangent and normal drawn at $P$ meet the major axis at points $T$ and $N$ respectively,then the value of $\frac{(\left| F_2N \right| + \left| F_1N \right|)(\left| F_2T \right| - \left| F_1T \right|)}{(\left| F_2N \right| - \left| F_1N \right|)(\left| F_2T \right| + \left| F_1T \right|)}$ is equal to (where $F_1$ and $F_2$ are the foci $(ae, 0)$ and $(-ae, 0)$ respectively).

$A$ man running round a race-course notes that the sum of the distances of two flag-posts from him is always $10 \ m$ and the distance between the flag-posts is $8 \ m$. The area of the path he encloses in square metres is (in $\pi$)