Demonstration for uniqueness of symmetry axis of parabola

Question

$$ P: (x)^2 = 2p(y) $$

is a parabola in $R²$.

How would I go about proving it has a single axis of symmetry?

Thank you for your time :)

Answer 1

An equivalent definition of axis of symmetry to yours is that it is a line such that the midpoint of every chord perpendicular to it lies on the line.

Given an arbitrary line $ax+by+c=0$ with $a$ and $b$ not both zero, we have the one-parameter family of its perpendiculars $-bx+ay+\lambda=0$. The midpoint of the chord that lies on this perpendicular can be found, among other ways, via pole-polar relationships: the polar of the midpoint of a chord is parallel to the chord and passes through the midpoint’s pole. Its coordinates work out to be $\left(\frac{pb}a,{pb^2-a\lambda\over a^2}\right)$. These points form the line $ax=pb$, so if there is an axis of symmetry, it must be parallel to the $y$-axis. Moreover, comparing this to the general equation $ax+by+c=0$ that we started with, we must have $b=0$, leaving only the line $x=0$ as an axis of symmetry of the parabola.

Note that in passing we’ve also proven another useful fact about parabolas: the midpoints of parallel chords lie on a line parallel to the parabola’s axis.