7. Let $AB$ be a chord through the center of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and $F$ be a focus. Then the maximum value of the area of $\triangle ABF$ is $bc\left(c^{2}=a^{2}-b^{2}\right)$.
7. Let $Q$ be the other focus, and connect $A Q, B Q$ to form the parallelogram $A F B Q$, then $S_{\triangle A B F}=S_{\triangle A F Q} \leqslant \frac{1}{2} \cdot 2 c \cdot b=b c$.