Find the curves for which the area of the triangle formed by…
Find the curves for which the area of the triangle formed by the tangent line, the ordinate of the point of tangency, and the $x$-axis is constant and equal to $a^2$.
Explanation of Method
The triangle has legs equal to the ordinate $|y|$ and the subtangent $|y/y'|$, so its area is $\tfrac12|y^2/y'|$. Setting this equal to $a^2$ gives two separable first-order equations to solve.
Step 1: Set up the ODE
\[\frac12\left|\frac{y^2}{y'}\right|=a^2 \quad\Longrightarrow\quad \frac{y^2}{y'}=\pm2a^2.\]
Step 2: Solve each case
Case $+$: $y'=\dfrac{y^2}{2a^2}$ separates as $\dfrac{dy}{y^2}=\dfrac{dx}{2a^2}$, integrating to $-\dfrac1y=\dfrac{x+C}{2a^2}$, i.e. $y=-\dfrac{2a^2}{x+C}$.
Case $-$: $y'=-\dfrac{y^2}{2a^2}$ similarly gives $y=\dfrac{2a^2}{x-C}$.
Result
Final answer:
\[\boxed{\,y=\pm\dfrac{2a^2}{x+C},\qquad C\in\mathbb{R}}\]
— a family of rectangular hyperbolas.