The foci of the ellipse $25(x + 1)^2 + 9(y + 2)^2 = 225$ are at

Prove that the product of the lengths of the perpendiculars drawn from the points $(\sqrt{a^{2}-b^{2}}, 0)$ and $(-\sqrt{a^{2}-b^{2}}, 0)$ to the line $\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$ is $b^{2}$.

The length of the latus rectum of the ellipse $5x^2 + 9y^2 = 45$ is

The equations of the directrices of the ellipse $9x^2 + 4y^2 - 18x - 16y - 11 = 0$ are

If $B$ is the end point of the minor axis of the ellipse $b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2}$ $(a > b)$ and $S$ and $S^{\prime}$ are the foci of the ellipse such that $\Delta SBS^{\prime}$ is an equilateral triangle,then the eccentricity $e$ is:

The centre of a circle $C$ is at the centre of the ellipse $E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four points. If the area of the triangle $PF_1F_2$ is $30$ and the length of the major axis of $E$ is $17$,then the distance between the foci of $E$ is: