Consider ellipses E_k: k x^2+k^2 y^2=1, k=1,2, …, 20. Let C_k be the circle which touches the four chords joining the end points (one on minor axis…

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