The value of $k$ such that the line $y=2x+k$ touches the ellipse $3x^2+5y^2=15$ is

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

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 a point $P(x, y)$ moves along the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and if $C$ is the centre of the ellipse,then the sum of maximum and minimum values of $CP$ is

The foci of the ellipse $9x^2 + 25y^2 = 225$ are

The equation $2x^2 + 3y^2 = 30$ represents