SEAMO 2027 - Paper E - SET 04 - Self-study Question 4

On the $xy$-plane, let $S$ consist of all points $(x,y)$ satisfying $\left|x+\frac12y\right|\le10$, $|x|\le10$ and $|y|\le10$. The largest circle centred at $(0,0)$ that fits inside $S$ has area $k\pi$. Find $k$.

Enter the number only (an exact fraction is allowed).