What is the standard form of the equation of the parabola with a directrix at x=-9 and a focus at (8,4)?

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Answer 1 · 2017-11-01T08:06:02

The equation of the parabola is $(y-4)^2=17(2x+1)$

Any point $(x,y)$ on the parabola is equidistant from the directrix and the focus.

Therefore,

$x-(-9)=sqrt((x-(8))^2+(y-(4))^2)$

$x+9=sqrt((x-8)^2+(y-4)^2)$

Squaring and developing the $(x-8)^2$ term and the LHS

$(x+9)^2=(x-8)^2+(y-4)^2$

$x^2+18x+81=x^2-16x+64+(y-4)^2$

$(y-4)^4=34x+17=17(2x+1)$

The equation of the parabola is $(y-4)^2=17(2x+1)$

graph{((y-4)^2-34x-17)((x-8)^2+(y-4)^2-0.05)(y-1000(x+9))=0 [-17.68, 4.83, -9.325, 1.925]}