The equation of normal at the point $(0, 3)$ of the ellipse $9{x^2} + 5{y^2} = 45$ is

The equations of the directrices of the ellipse $16{x^2} + 25{y^2} = 400$ are

If the points of intersection of two distinct conics $x^2+y^2=4 b$ and $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ lie on the curve $y^2=3 x^2$, then $3 \sqrt{3}$ times the area of the rectangle formed by the intersection points is............................

Find the equation of the ellipse, whose length of the major axis is $20$ and foci are $(0,\,\pm 5)$

On the ellipse $4{x^2} + 9{y^2} = 1$, the points at which the tangents are parallel to the line $8x = 9y$ are

Consider ellipses $E _{ k }: kx ^2+ k ^2 y ^2=1, k =1,2, \ldots$,$20$. Let $C _{ k }$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$, If $r_k$ is the radius of the circle $C _{ k }$, then the value of $\sum \limits_{ k =1}^{20} \frac{1}{ I _{ k }^2}$ is $.......$.