1. In rectangle $A B C D$, $A B=a, B C=b, M$ is the midpoint of $B C$, $D E \perp$ ray $A M$ at $E$, find the length of $D E$. (Express in terms of $a, b$)
1. Extend $A M$ to intersect the extension of $D C$ at $F$, and draw $F G / / B C$ to intersect the extension of $A B$ at $G$, then $B G=a, G F=b$. Taking $D$ as the viewpoint, apply the Angle Bisector Theorem to $A, E, F$, we have $\frac{\sin 90^{\circ}}{D E}=\frac{\sin \alpha}{D F}+\frac{\sin \beta}{D A}$, which simplifies to $\frac{1}{D E}=\frac{\sin \alpha}{2 a}+\frac{\sin \beta}{b}$. In the right triangle $\triangle A D F$, $\sin \alpha=$ $\frac{b}{\sqrt{4 a^{2}+b^{2}}}, \sin \beta=\frac{2 a}{\sqrt{4 a^{2}+b^{2}}}$. Therefore, we find $D E=\frac{2 a b \sqrt{4 a^{2}+b^{2}}}{4 a^{2}+b^{2}}$.