HMMT-Nov 2014 gen, Problem 1

Two circles $\omega$ and $\gamma$ have radii 3 and 4 respectively, and their centers are 10 units apart. Let $x$ be the shortest possible distance between a point on $\omega$ and a point on $\gamma$, and let $y$ be the longest possible distance between a point on $\omega$ and a point on $\gamma$. Find the product $x y$.

Solution: Let $\ell$ be the line connecting the centers of $\omega$ and $\gamma$. Let $A$ and $B$ be the intersections of $\ell$ with $\omega$, and let $C$ and $D$ be the intersections of $\ell$ with $\gamma$, so that $A, B, C$, and $D$ are collinear, in that order. The shortest distance between a point on $\omega$ and a point on $\gamma$ is $B C=3$. The longest distance is $A D=3+10+4=17$. The product is 51 . $\fbox{51}$.

Answer: 51