The area of the greatest rectangle that can be inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is

In an ellipse,two vertices are $(5,0)$ and $(0,-4)$. Then the equation of the ellipse is

The equation to the locus of the middle point of the portion of the tangent to the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ included between the coordinate axes is:

Find the length of the latus rectum of the ellipse $4x^2 + 9y^2 - 36y + 4 = 0$.

The equation of an ellipse whose eccentricity is $1/2$ and the vertices are $(4, 0)$ and $(10, 0)$ is

Consider ellipses $E_{k}: kx^{2} + k^{2}y^{2} = 1$,for $k = 1, 2, \ldots, 20$. Let $C_{k}$ be the circle which touches the four chords joining the end points (one on the minor axis and another on the major axis) of the ellipse $E_{k}$. If $r_{k}$ is the radius of the circle $C_{k}$,then the value of $\sum_{k=1}^{20} \frac{1}{r_{k}^{2}}$ is $.......$.