If $OB$ is the semi-minor axis of an ellipse,$F_1$ and $F_2$ are its foci and the angle between $F_1B$ and $F_2B$ is a right angle,then the square of the eccentricity of the ellipse is

The radius of the circle passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and having its center at $(0, 3)$ is:

Tangents are drawn to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$ at the ends of both latus recta. The area of the quadrilateral so formed is

The equation of the ellipse whose centre is at the origin and which passes through the points $(-3, 1)$ and $(2, -2)$ is

If the focus of an ellipse is $(-1, -1)$,the equation of its directrix corresponding to this focus is $x + y + 1 = 0$,and its eccentricity is $e = \frac{1}{\sqrt{2}}$,then the length of its major axis is

$A$ staircase of length $l$ rests against a vertical wall and a floor of a room. Let $P$ be a point on the staircase,nearer to its end on the wall,that divides its length in the ratio $1 : 2$. If the staircase begins to slide on the floor,then the locus of $P$ is