2. The segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 4 and 1. Find the common chord, given that the radii of the circles are in the ratio $3: 2$.

Answer. $2 \sqrt{11}$.
Let 0 be the center of the circle with radius $2R$, $Q$ be the center of the circle with radius $3R$; $AB$ be the common chord of the circles; $A B \cap 0 Q=N$.

Since $A N \perp O Q$, we have $A N^{2}=4 R^{2}-1=9 R^{2}-16$, from which we find that $R^{2}=3$. Therefore, $A N^{2}=11$, $A B=2 \cdot A N=2 \sqrt{11}$.