Let $P(2,2)$ be a point on an ellipse whose foci are $F_{1}(5,2)$ and $F_{2}(2,6)$. Find the eccentricity of the ellipse.

Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be equal to the length of its semi-major axis. If the radius of its director circle is $\sqrt{3}$ and $e$ is its eccentricity,then the length of its latus rectum is

If the foci and vertices of an ellipse are $(\pm 1, 0)$ and $(\pm 2, 0)$ respectively,then the length of the minor axis of the ellipse is:

If tangents are drawn to the ellipse $x^2+2y^2=2$,then the locus of the midpoints of the intercepts made by the tangents between the coordinate axes is

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

If $A$ and $B$ are two fixed points and $P$ is a variable point such that $PA + PB = 4$,then the locus of $P$ is a/an