2. The segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the common chord, given that the radii of the circles are in the ratio $4: 3$.

Answer. $2 \sqrt{23}$.
Let 0 be the center of the circle with radius $3 R$, $Q$ be the center of the circle with radius $4 R$; $AB$ be the common chord of the circles; $A B \cap O Q=N$.

Since $A N \perp 0 Q$, we have $A N^{2}=9 R^{2}-4=16 R^{2}-25$, from which we find that $R^{2}=3$. Therefore, $A N^{2}=23$, $A B=2 \cdot A N=2 \sqrt{23}$.