We know that $\frac{e}{f}=\frac{3}{4}$ and $\sqrt{e^{2}+f^{2}}=15$. Solving for $e$ and $f$, we find that $e^{2}+f^{2}=225$, so $16 e^{2}+16 f^{2}=3600$, so $(4 e)^{2}+(4 f)^{2}=3600$, so $(3 f)^{2}+(4 f)^{2}=3600$, so $f^{2}\left(3^{2}+4^{2}\right)=3600$, so $25 f^{2}=3600$, so $f^{2}=144$ and $f=12$. Thus, $e=\frac{3}{4} \cdot 12=9$. Therefore, $\boldsymbol{e f}=9 * 12=\mathbf{1 0 8}$.