$P$ is a variable point on the parabola $x^2 + 44x = y + 88$ and $Q$ is a point on the plane not lying on the parabola if $(PQ)^2$ is minimum, then?

Question

Full question is : $P$ is a variable point on the parabola $x^2 + 44x = y + 88$ and $Q$ is a point on the plane not lying on the parabola if $(PQ)^2$ is minimum, then the angle between tangent at $P$ and $PQ$ is ?

I tried to solve by taking a variable point $(h, h^2 + 44h -88 )$ on the parabola and a point $(x,y)$ outside it and by using distance formula and then trying to find minimum distance by maximima minima , but it is going very lengthy and is not useful in finding the answer asked in the question .

Answer 1

The first thing I would do is complete the square: $$y= x^2+ 44x- 88= (x^2+ 44x+ 484)- 398= (x+ 22)^2- 398$$

So that the translation $x'= x+ 22$ and $y'= y+ 398$ changes the equation to $y'= x'^2$. It should be clear that a translation does not change the angles at all so now prove the same thing about this parabola.

Answer 2

The angle between PQ and tangent at P is always $90^0$ ! ... because it should be normal to the parabola for minimal distance.