If the curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{25}+\frac{y^2}{16}=1$ cut each other orthogonally, then $a^2-b^2$ equals to

Let $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ be an ellipse. Ellipses $E_i$ are constructed such that their centres and eccentricities are the same as that of $E_1$,and the length of the minor axis of $E_i$ is the length of the major axis of $E_{i+1}$ $(i \geq 1)$. If $A_i$ is the area of the ellipse $E_i$,then $\frac{5}{\pi}\left(\sum_{i=1}^{\infty} A_i\right)$ is equal to . . . . . . .

The number of values of $c$ such that the line $y=4x+c$ touches the curve $\frac{x^{2}}{4}+y^{2}=1$ is

$A$ line passing through the endpoint of the major axis $A$ and the endpoint of the minor axis $B$ of an ellipse $\frac{x^2}{9} + y^2 = 1$ touches its auxiliary circle at point $M$. Find the area of the triangle with vertices $A, M,$ and the origin $O$.

Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 5, 0)$,foci $(\pm 4, 0)$.

The ellipse having its foci $(0, \pm 1)$ and major axis of length $\sqrt{5}$ is