If $P$ is a point on $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with foci $S$ and $S^{\prime}$,then the maximum area of $\triangle S P S^{\prime}$ is

Let $(h, k)$ lie on the circle $C: x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to . . . . . . .

If $x+y+n=0, n>0$ is a normal to the ellipse $x^2+3y^2=3$ and $x+my+3=0, m < 0$ is a tangent to the ellipse $x^2+5y^2=5$,then the point of intersection of these two lines satisfies the equation

Let the ellipse $E : x^2 + 9y^2 = 9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A, P$ and the origin $O$ is $\frac{m}{n}$,where $m$ and $n$ are coprime,then $m - n$ is equal to

Find the equation for the ellipse that satisfies the given conditions: $b=3, c=4,$ centre at the origin; foci on the $x$-axis.

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 . . . . . . .