2. In $\triangle A B C$, $A B=4, C A: C B=5: 3$. Try to find $\left(S_{\triangle A B C}\right)_{\max }$.
Construct the Apollonian circle $\odot O$ of $AB$ with the ratio $5:3$, and find the point $C$ on $\odot O$ that is farthest from $AB$. It is easy to see that the diameter $MN=7.5$ of $\odot O$. Draw the radius $OC \perp MN$ through $O$. Then $OC=3.75$. Therefore, $\left(S_{\triangle ABC}\right)_{\max }=7.5$.