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

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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=(a^2)/(52) \end{gathered}

So, the equation is:


y=(x^2)/(52)

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