Describing all the plane regular curves whose tangent/normal lines all intersect at a fixed point

Question

My intuition tells me that the first case (tangent lines all passing through a fixed point) corresponds to a segment of a line, and the second one corresponds to a circle, but I'm having trouble actually proving it. I thought about assuming (without loss of generality) that the fixed point is the origin, but I didn't get anywhere. For the second one, it would be enough to prove that it has constant curvature... but I'm not seeing where to start.

Answer 1

Suppose $P = (x_0, y_0)$ is a point on the first curve. The tangent $L$ at $P$ is given by the equation \begin{align*} y - y_0 = y'(x_0) (x - x_0). \end{align*}

If $L$ passes through the origin, then $y_0 - x_0 y'(x_0) = 0$. This is a differential equation that can be solved by separation of variables. It gives the equation of a line.

The second curve can be found by following the same method.