14. On the $x y$-plane, let $S$ denote the region consisting of all points $(x, y)$ for which
$$
\left|x+\frac{1}{2} y\right| \leq 10, \quad|x| \leq 10 \text { and } \quad|y| \leq 10 .
$$

The largest circle centred at $(0,0)$ that can be fitted in the region $S$ has area $k \pi$. Find the value of $k$.
14. Answer: 80 .
The region $S$ is the hexagon enclosed by the lines
$$
x= \pm 10, \quad y= \pm 10, \quad x+\frac{1}{2} y= \pm 10 .
$$

The largest circle contained in $S$ is tangent to $x+\frac{1}{2} y= \pm 10$. Hence, its radius is the distance from the origin $(0,0)$ to $x+\frac{1}{2} y=10$ :
$$
r=\frac{10}{\sqrt{1+(1 / 2)^{2}}}=4 \sqrt{5} \text {. }
$$

The aree of the largest circle is thus $\pi r^{2}=\pi(4 \sqrt{5})^{2}=80 \pi$.