Question

find the equation of a parabola with a focus of (0, 13) and directrix y = -13

Answer 1

If the focus of the parabola is (0,13) and the directrix is y=-13.

Let any point on the parabola = (a,b).

First, we find the distance between (a,b) and the focus (0, 13).

$\begin{gathered} \text{Distance}=\sqrt[]{(a-0)^2+(b-13)^2} \\ =\sqrt[]{a^2+(b-13)^2} \end{gathered}$

Next, find the distance between (a,b) and the directrix y=-13.

$|b-(-13)|=|b+13|$

Next, equate the two expressions obtained.

$\sqrt[]{a^2+(b-13)^2}=|b+13|$

Square both sides.

$\begin{gathered} a^2+(b-13)^2=(b+13)^2 \\ a^2+b^2-26b+169=b^2+26b+169 \\ a^2+b^2-b^2-26b-26b+169-169=^{}0 \\ a^2-52b=0 \\ 52b=a^2 \\ b=\frac{a^2}{52} \end{gathered}$

So, the equation is:

$y=\frac{x^2}{52}$