Find the coordinates of the foci of the ellipse $25(x + 1)^2 + 9(y + 2)^2 = 225$.

If $A$ and $B$ are two fixed points and $P$ is a variable point such that $PA + PB = 4$,then the locus of $P$ is a/an

Point $O$ is the centre of the ellipse with major axis $AB$ and minor axis $CD$. Point $F$ is one focus of the ellipse. If $OF = 6$ and the diameter of the inscribed circle of triangle $OCF$ is $2$,then the product $(AB)(CD)$ is equal to

If the latus rectum of an ellipse subtends a right angle at the centre of that ellipse,then the eccentricity of that ellipse is

$A$ tangent is drawn to the ellipse $\frac{x^2}{32} + \frac{y^2}{8} = 1$ from the point $A(8, 0)$ to touch the ellipse at point $P$. If the normal at $P$ meets the major axis of the ellipse at point $B$,then the length $BC$ is equal to (where $C$ is the center of the ellipse) - ............ $units$

$A$ tangent is drawn at $(3 \sqrt{3} \cos \theta, \sin \theta)$ $\left(0 < \theta < \frac{\pi}{2}\right)$ to the ellipse $\frac{x^2}{27}+\frac{y^2}{1}=1$. The value of $\theta$ for which the sum of the intercepts on the coordinate axes made by this tangent attains the minimum is