The equation of family of curves for which the length of the normal is equal to the radius vector, is

We have,

Length of the normal $=y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$

It is given that

$\begin{array}{l}y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}=\sqrt{x^2+y^2}\left[\because \text{ Radius vector }=R=\sqrt{x^2+y^2}\right]\\ \Rightarrow y^2+y^2{\left(\frac{dy}{dx}\right)}^2=x^2+y^2\\ \Rightarrow y^2{\left(\frac{dy}{dx}\right)}^2=x^2\Rightarrow ydy\pm xdx=0\Rightarrow y^2\pm x^2=k^2\end{array}$