Find the family of curves for which the length of the…
Find the family of curves for which the length of the tangent segment (from the point of tangency to the $x$-axis) equals the distance from the point of tangency to the origin.
Explanation of Method
Write both lengths in terms of $x$, $y$, $y'$, equate and square to clear the radicals; the condition collapses remarkably to $y'=\pm y/x$, giving two elementary families.
Step 1: Set up the equation
The tangent-segment length is $\dfrac{|y|\sqrt{1+(y')^2}}{|y'|}$ and the distance to the origin is $\sqrt{x^2+y^2}$. Equating and squaring,
\[\frac{y^2(1+(y')^2)}{(y')^2}=x^2+y^2 \ \Longrightarrow\ y^2=(y')^2x^2 \ \Longrightarrow\ y'=\pm\frac{y}{x}.\]
Step 2: Solve both cases
$y'=\dfrac{y}{x}$ separates to $\dfrac{dy}{y}=\dfrac{dx}{x}$, giving $y=Cx$ (lines through the origin).
$y'=-\dfrac{y}{x}$ gives $\dfrac{dy}{y}=-\dfrac{dx}{x}$, i.e. $\ln|xy|=$ const, so $xy=C$ (rectangular hyperbolas).
Result
Final answer:
\[\boxed{\,y=Cx\qquad\text{or}\qquad xy=C.}\]