Question

What is the equation of a parabola with (4, 6) as its focus and y = 2 as its directrix?

Answer 1

So  hmmm if you notice the picture below

bear in mind that, the vertex is "p" distance from the focus, and "p" distance from the directrix

now, the focus is at 4,6 and the directrix below that, meaning the parabola is vertical and opening upwards

and the vertex is half-way between the focus point and the directrix

$\bf \textit{parabola vertex form with focus point distance}\\\\ \begin{array}{llll} (y-{{ k}})^2=4{{ p}}(x-{{ h}}) \\\\ \boxed{(x-{{ h}})^2=4{{ p}}(y-{{ k}})} \\ \end{array} \qquad \begin{array}{llll} vertex\ ({{ h}},{{ k}})\\\\ {{ p}}=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix} \end{array}\\\\ -----------------------------\\\\ (x-4)^2=4(2)(y-4)\implies (x-4)^2=8(y-4) \\\\\\ \cfrac{(x-4)^2}{8}=y-4\implies \cfrac{1}{8}(x-4)^2+4=y$