The minimum area of a triangle formed by any tangent to the ellipse $\frac{x^2}{16} + \frac{y^2}{81} = 1$ and the coordinate axes is

If the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ having $(1,1)$ as its midpoint is $x+\alpha y=\beta$,then

The product of the lengths of perpendiculars from the foci on any tangent to the ellipse $3x^2 + 5y^2 = 1$ is:

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

For real numbers $a, b$ $(a > b > 0)$,let $\text{Area} \{(x, y) : x^{2} + y^{2} \leq a^{2} \text{ and } \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} \geq 1\} = 30\pi$ and $\text{Area} \{(x, y) : x^{2} + y^{2} \geq b^{2} \text{ and } \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} \leq 1\} = 18\pi$. Then the value of $(a - b)^{2}$ is equal to

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 $.......$.