38.4. Find the envelope of the family of lines that cut off from a given right angle a triangle of area $a^{2} / 2$.
38.4. Let's introduce a coordinate system, directing the axes along the sides of the given right angle. The lines of interest intersect the coordinate axes at points ( $\alpha a, 0$ ) and ( $0, a / \alpha$ ), where $\alpha>0$. The line corresponding to the parameter $\alpha$ is given by the equation $x+\alpha^{2} y=\alpha a$. Two lines corresponding to parameters $\alpha_{1}$ and $\alpha_{2}$ intersect at the point with coordinates $\left(\frac{a \alpha_{1} \alpha_{2}}{\alpha_{1}+\alpha_{2}}, \frac{a}{\alpha_{1}+\alpha_{2}}\right)$. If $\alpha_{1} \rightarrow \alpha$ and $\alpha_{2} \rightarrow \alpha$, we obtain the point $\left(\frac{a \alpha}{2}, \frac{a}{2 \alpha}\right)$. Such points lie on the hyperbola $x y=a^{2} / 4$.