P(2ap,ap^2) is a variable point on the parabola x^2=4ay, where the focus is S.

Question

$P(2ap,ap^2)$ is a variable point on the parabola $x^2=4ay$, where the focus is $S$. $Q$ divides the interval from $P$ to $S$ in the ratio $p^2:1$. Where $x=2ap/p^2+1$ and $y=2ap^2/p^2+1$

1. Show that $y/x=p$
2. Prove that as $P$ moves, $Q$ moves in a circle and state the centre of that circle.

I can do part 1 but I am not sure how to $p$ and $Q$ move in circle in part 2

Thanks!

Answer 1

Here’s a way to proceed given that you already know the locus is supposed to be a circle.

An equation of a circle is of the form $x^2+y^2+\dots=0$, where the elision represents linear and constant terms, so try computing $$\begin{align} x^2+y^2 &= {4a^2p^2\over(p^2+1)^2}+{4a^2p^4\over(p^2+1)^2} \\ &= {4a^2p^2\over(p^2+1)} \\ &= 2ay.\end{align}$$ Rearranging and completing the square produces $$x^2+(y-a)^2=a^2,$$ the equation of a circle of radius $a$ centered at $(0,a)$.