Let the eccentricity of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b$, be $\frac{1}{4}$. If this ellipse passes through the point $\left(-4 \sqrt{\frac{2}{5}}, 3\right)$, then $a^{2}+b^{2}$ is equal to

The equation of the ellipse whose one of the vertices is $(0,7)$ and the corresponding directrix is $y = 12$, is

The centre of the ellipse$\frac{{{{(x + y - 2)}^2}}}{9} + \frac{{{{(x - y)}^2}}}{{16}} = 1$ is

Tangent is drawn to ellipse $\frac{{{x^2}}}{{27}} + {y^2} = 1$ at $(3\sqrt 3 \cos \theta ,\;\sin \theta )$ where $\theta \in (0,\;\pi /2)$. Then the value of $\theta $ such that sum of intercepts on axes made by this tangent is minimum, is

Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let the line passing through $P$ and parallel to $y$-axis meet the circle $x^2+y^2=9$ at point $Q$ such that $P$ and $Q$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $P Q$ such that $P R: R Q=4: 3$ as $P$ moves on the ellipse, is :

Area (in sq. units) of the region outside $\frac{|\mathrm{x}|}{2}+\frac{|\mathrm{y}|}{3}=1$ and inside the ellipse $\frac{\mathrm{x}^{2}}{4}+\frac{\mathrm{y}^{2}}{9}=1$ is