Given a parabola $y^2 = x$, **how can we find the equation of the curve formed by the intersection of normals drawn from different points?**
I have attached an image for reference.
[![](https://i.stack.imgur.com/UYb4S.jpg)](https://i.stack.imgur.com/UYb4S.jpg)
I tried taking two close points $x$ and $x+dx$ and tried finding the intersection of their normals and solving the differential equation to get the answer but I couldn't get it right. Could someone explain how to find this either using differential equations or by any other method.