[Math] Locus of centre of variable circle

Question

I am not able to figure out this question

What is the locus of the centre of a circle which touches a given line and passes through a given point, not lying on the given line?

I think it's a parabola but I am not able to prove it mathematically

Answer 1

Without loss of generality, we may assume that the line is the $x$-axis, and that the point is the point $(0,a)$, where $a$ is positive.

A circle with centre $(p,q)$ that touches the $x$-axis must have radius $|q|$. So it has equation $$(x-p)^2+(y-q)^2=q^2.$$ The circle goes through $(0,a)$. It follows that $$(0-p)^2+(a-q)^2=q^2.$$ Simplify. We get $$p^2+a^2-2aq=0.$$ We can rewrite this as $$q=\frac{1}{2a}p^2 +\frac{a}{2},$$ indeed the equation of a parabola.