Russ P. answered • 11/11/14

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Casey,

The defining characteristic of a parabola is that any (x,y) point on it is exactly the same distance away from its focus point and the point on the directrix that forms a perpendicular line segment to the focus point.

The distance from the focus point (3,1) is √[(x-3)

^{2}+ (y-1)^{2}].And the perpendicular distance from (3,1) to the x=9 directrix line point (9,1) is just |x-9| since Δy = 0.

Hence, solve: √[(x-3)

^{2}+ (y-1)^{2}] = |x-9| by squaring both sides, expanding, and combining like terms:[(x-3)

^{2}+ (y-1)^{2}] = (x-9)^{2}so (x^{2}- 6x +9) + (y^{2}- 2y + 1) = x^{2}-18x + 81 , the x^{2}terms cancel out, andx = (-1/12) y

^{2}+ (1/6) y + (71/12)That's the equation for this parabola. If you graph it, it opens up to the left, is symmetrical about the line y=+1 through the focus point, and goes thru (6, 1) where its distance is 3 from the focus and the directrix line.