Coordinate Geometry: JEE Main 2023 Mathematics question

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