To 15(Ei). Draw a chord of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ passing through B $(0,-b)$. Find the maximum chord length.
Let $(\mathrm{x}, \mathrm{y})$ be a point on the ellipse, then $|\mathrm{BM}|^{2}$ $=x^{2}+(y+b)^{2}$. Substituting $x^{2}=\frac{a^{2}}{b^{2}}\left(b^{2}-y^{2}\right)$, we get: $|B M|^{2}=\left(1-\frac{a^{2}}{b^{2}}\right) y^{2}+2 b y+a^{2}$ $+b^{2}$. Since $1-\frac{a^{2}}{b^{2}}<0$, the maximum value of $|B M|^{2}$ is:
$$
|B M|^{2} \max =\frac{4\left(1-\frac{a^{2}}{b^{2}}\right)\left(a^{2}+b^{2}\right)-4 b^{2}}{4\left(1-\frac{a^{2}}{b^{2}}\right)}
$$