Let the length of a latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be $10$. If its eccentricity is the minimum value of the function $f(t) = t^2 + t + \frac{11}{12}$,$t \in R$,then $a^2 + b^2$ is equal to:

If $CP$ and $CD$ are a pair of semi-conjugate diameters of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$,then $CP^{2}+CD^{2}=$

If the eccentricity of an ellipse is $5/8$ and the distance between its foci is $10$,then its latus rectum is

Find the coordinates of the foci,the vertices,the length of the major axis,the minor axis,the eccentricity,and the length of the latus rectum of the ellipse $36 x^{2}+4 y^{2}=144$.

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

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be an ellipse with foci $F_1$ and $F_2$. Let $AO$ be its semi-minor axis,where $O$ is the centre of the ellipse. The lines $AF_1$ and $AF_2$,when extended,cut the ellipse again at points $B$ and $C$ respectively. Suppose that the $\triangle ABC$ is equilateral. Then,the eccentricity of the ellipse is