If the distance between the foci of an ellipse is half the length of its latus rectum,then the eccentricity of the ellipse is

Find the number of real tangents that can be drawn from the point $(3, 5)$ to the ellipses $3x^2 + 5y^2 = 32$ and $25x^2 + 9y^2 = 450$.

The area (in square units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\frac{x^2}{9} + \frac{y^2}{5} = 1$ is:

The distance between the directrices of the conic $\sqrt{(x - 5)^2 + (y - 4)^2} + \sqrt{(x - 3)^2 + (y - 2)^2} = 6$ is

$P$ is a point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. When the area of $\Delta PSS'$ is maximum,the inradius of $\Delta PSS'$ ($S$ and $S'$ are foci) is equal to:

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $E: \frac{x^2}{16} + \frac{y^2}{9} = 1$ at the points $A$ and $B$. Let the line $y = x$ intersect $E$ at the points $C$ and $D$. Then the area of the quadrilateral $ABCD$ is equal to