Ellipse

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 :