Find the equation of the ellipse whose vertices are $(\pm 13, 0)$ and foci are $(\pm 5, 0)$.

The area of the region bounded by the curve $x = 4 \cos \theta, y = 3 \sin \theta$ is . . . . . . sq. units. (in $\pi$)

If the length and breadth of a rectangle of maximum area that can be inscribed in an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $8 \sqrt{2}$ and $4 \sqrt{2}$ respectively,then the eccentricity of that ellipse is

The eccentricity of the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$ is

Consider the parabola $P : y^2 = 4x$ and the ellipse $E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of $P$ and $E$ be their common latus rectum. If the eccentricity of $E$ is $e$,then $e^2 + 2\sqrt{2}$ is equal to . . . . . .

Find the equation of the ellipse which passes through the points $(-3, 1)$ and $(2, -2)$,whose center lies at $(0, 0)$ and major axis lies along the $X$-axis.