The directrix of a parabola is y=9. the focus of the parabola is (2,5). what is the equation of the parabola?

The distance from any point P (x,y) on the parabola to the focus = the distance of this point from the directrix.

distance between P to focus:-

= sqrt ( x - 2)^2 + (y - 5))^2

distance from the point to directrix = ( 9 - y)

so sqrt( x - 2)^2 + (y-5)^2  = 9 -y
(x - 2)^2+ (y - 5)^2 = (9-y)^2
x^2 -4x + 4 + y^2 - 10y + 25 = 81 - 18y + y^2
x^2 - 4x + 4 +25 - 81 = -18y + 10y
-8y = x^2 - 4x -52 

y =  (-1/8)^2 + 1/2 x + 6 1/2    is the required equation

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