Is the parabola the only differentiable convex even curve, such that all vertical lines, reflected on the tangent of the curve, converge into a point?

Question

• It is well known in optics that a parabola $y = x^2$ has this very important property for applications:

  $(P)$ The reflection of all vertical lines on the tangent of the parabola all converge into a single point

  See Formula for the equation of the reflection of a line on another line, and application to convergence of reflections of vertical lines on a parabola.
  See also image 1 below.

• It is not true for other curves, for example for a spherical function $y = -\sqrt{1-x^2}+1$, see image 2.

Question: is it true that the only differentiable, even, convex, functions satisfying $(P)$ are the parabolas?

See also this about parabolas: