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I have posted a link there to this thread so the OP can view my work.Writing an equation given the directrix and focus?

focus: (3,5) directrix y=1. write an equation for the parabola. How do I do this? please help!!

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I have posted a link there to this thread so the OP can view my work.Writing an equation given the directrix and focus?

focus: (3,5) directrix y=1. write an equation for the parabola. How do I do this? please help!!

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Hello jose,

Let's let $(x,y)$ be an arbitrary point on the parabola. Now, we know the perpendicular distance from the point to the directrix will be equal to the distance between this point and the focus. Thus, we may state:

\(\displaystyle |y-1|=\sqrt{(x-3)^2+(y-5)^2}\)

Square both sides:

\(\displaystyle (y-1)^2=(x-3)^2+(y-5)^2\)

Expand binomials:

\(\displaystyle y^2-2y+1=x^2-6x+9+y^2-10y+25\)

Collect like terms:

\(\displaystyle 8y=x^2-6x+33\)

Divide through by $8$:

\(\displaystyle y=\frac{x^2-6x+33}{8}\)