If the eccentric angles of the extremities of a focal chord (other than the major axis) of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ are $\alpha$ and $\beta$,then $\frac{\cot(\alpha/2)}{\tan(\beta/2)}=$

The centre of the ellipse $4x^2 + 9y^2 - 16x - 54y + 61 = 0$ is

The eccentricity of the ellipse $(x-3)^2 + (y-4)^2 = \frac{y^2}{9} + 16$ is -

Find the equation for the ellipse that satisfies the given conditions: Vertices $(0, \pm 13)$,foci $(0, \pm 5)$.

If $S$ and $S^{\prime}$ are the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and if $PSP^{\prime}$ is a focal chord with $SP=8$,then $SS^{\prime}$ is equal to

Let $f$ be a strictly decreasing function defined on $\mathbb{R}$ such that $f(x) > 0, \forall x \in \mathbb{R}$. Let $\frac{x^2}{f(a^2+5a+3)} + \frac{y^2}{f(a+15)} = 1$ be an ellipse with the major axis along the $y$-axis. The value of $a$ can lie in the interval$(s)$: