Why is the focus of the parabola not within the parabola in the following result? So i'm going through my book and try to solve the following question:
Find the equation of the parabola which is symmetric about the y axis and passed through the point (2,-3).
Since it passes through (2,-3), we can assume that the parabola opens downwards, and hence use the equation $x^2 = -4ay$.
Plugging in the values though you'd get $4 = -4(-3)a$ or $a = \frac13$
But this implies that the focus is at $(0, \frac13)$ which is clearly not in the parabola. How is this possible/where did i go wrong?

You are confusing yourself here! If a parabola has equation $x^2=4ay$, then its focus is at $(0,a)$. But you have given your parabola the equation $x^2=-4ay$, so its focus is at $(0,-a)$.
Simpler, I would say, to stick with $x^2=4ay$ and let $a$ be negative.
