Find a projective change of coordinates

Question

Find a projective change of coordinates that takes the projective completion of the circumference C: $x^2 + y^2 = 1$ to the projective completion of the parabola P: $y^2=2px$, $p \geq 0$

(i.e. $x^2 + y^2 = z^2$ to $y^2=2pxz$)

Answer 1

This answer I found myself works as my professor checked it. The answer is in Spanish but I feel it's pretty easy to understand.