Consider ellipses \( E_k : kx^2 + k^2y^2 = 1, \, k = 1, 2, \ldots, 20 \). 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 \( \displaystyle \sum_{k=1}^{20} \frac{1}{r_k^2} \) is:
