SOLUTION: find the equation of the parabola with the given focus and directrix. focus ​(​5,-2​), directrix y=1

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Question 1162872: find the equation of the parabola with the given focus and directrix. focus ​(​5,-2​), directrix y=1
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Use the definition for Parabola based on the Distance Formula.

All points (x,y) equally distant from (5,-2) and (x,1).

%28x-5%29%5E2%2B%28y%2B2%29%5E2=%28x-x%29%5E2%2B%28y-1%29%5E2, and simplify from here,...
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If no mistakes made done on paper, then 6y%2B3=-%28x-5%29%5E2.

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The focus is below the directrix, so the parabola opens downward. The vertex form of the equation of the parabola is

y-k+=+%281%2F%284p%29%29%28x-h%29%5E2

where (h,k) is the vertex and p is the directed distance from the vertex to the focus.

The vertex is halfway between the focus and the directrix: (5,-.5). That makes p = -1.5.

The equation is

y%2B.5+=%281%2F-6%29%28x-5%29%5E2
6y%2B3+=+-%28x-5%29%5E2