Are normal lines of a graph, tangent lines to another graph? And how can you induce it?

Question

I was interested into normal lines of a curve, so tried to find out how many normal lines I can draw if I chose a random dot on the coordinate plane. The normal line trace of $y=x^2$ turned out like this;

I want to find out the curve which the normal lines seem to become a tangent line of that curve, as you can see in the image. I also want to find out how to induce the function.