Consider ellipses E_k:kx^2+k^2y^2=1, k=1,2,,20.

Question

Consider ellipses $E_k:kx^2+k^2y^2=1$, $k=1,2,\dots,20$. Let $C_k$ be the circle which touches the four chords joining the endpoints (one on the minor axis, one on the major axis) of $E_k$. If $r_k$ is the radius of $C_k$, then the value of $\displaystyle\sum_{k=1}^{20}\frac1{r_k^2}$ is:

Accepted Answer

$E_k:\dfrac{x^2}{1/k}+\dfrac{y^2}{1/k^2}=1$, with axis endpoints $\left(\frac1{\sqrt k},0\right)$ and $\left(0,\frac1k\right)$. The chord joining them: $\sqrt k\,x+ky=1$, at distance
$$r_k=\frac1{\sqrt{k+k^2}}\ \Longrightarrow\ \frac1{r_k^2}=k+k^2$$
from the origin.

$$\sum_{k=1}^{20}\frac1{r_k^2}=\sum_{k=1}^{20}k+\sum_{k=1}^{20}k^2=210+2870=3080.$$

Solutions · 1