If the eccentricity of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b)$ is $e = 2/3$ and its focal chord is $3x + 2y - 6 = 0$, then the value of $a^2 + b^2$ is...

If the distance between the foci of an ellipse is equal to its minor axis,then its eccentricity is

How many real tangents can be drawn to the ellipse $5x^2 + 9y^2 = 32$ from the point $(2, 3)$?

Find the locus of the midpoint of the portion of any tangent to the ellipse $x^{2} + 2y^{2} = 2$ intercepted between the axes.

If the normal at one end of a latus rectum of the ellipse $\frac{x^2}{32}+\frac{y^2}{b^2}=1$ passes through one end of the minor axis,then $\frac{e^4}{1-e^2}=$ (Here $e$ is the eccentricity of the ellipse)

$A$ 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