Consider ellipses $E_k: k x^2+k^2 y^2=1, k=1,2, \ldots \ldots, 20$. 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 $E_{k_1}$ If $r_k$ is the radius of the circle $C_k$, then the value of $\sum_{k=1}^{20} \mathrm{r}_k^{\frac{1}{2}}$ is