Find the curves for which the distance from the tangent line…

Find the curves for which the distance from the tangent line to the origin equals the $x$-coordinate of the point of tangency.

Step 1: Set up the equation

The tangent line $y'X-Y+(y-xy')=0$ has distance $\dfrac{|y-xy'|}{\sqrt{1+(y')^2}}$ from the origin. Setting this equal to $x$ and squaring,

\[(y-xy')^2=x^2[1+(y')^2] \quad\Longrightarrow\quad y^2-2xyy'=x^2 \quad\Longrightarrow\quad y'=\frac{y^2-x^2}{2xy}.\]

Step 2: Substitute $y=vx$

\[x\frac{dv}{dx}=\frac{v^2-1}{2v}-v=-\frac{v^2+1}{2v}.\]

Step 3: Separate and integrate

\[\frac{2v\,dv}{v^2+1}=-\frac{dx}{x} \quad\Longrightarrow\quad \ln(v^2+1)=-\ln|x|+C \quad\Longrightarrow\quad x(v^2+1)=K.\]

Step 4: Back-substitute $v=y/x$

\[\boxed{\,x^2+y^2=Cx\ \ \Longleftrightarrow\ \ \left(x-\tfrac{C}{2}\right)^2+y^2=\tfrac{C^2}{4}}\]

— a family of circles through the origin, centered on the $x$-axis.