How do you write an equation for the parabola with focus (-4,0) and directrix x=6?

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Answer 1 · 2016-11-25T18:28:39

The equation is #y^2=-20(x-1)#

The vertex is at #((6+(-4))/2,0)=(1,0)#

The distance of any point #(x,y)# on the parabola is equal to the distance to the directrix and to the focus.

Therefore,

#sqrt(x-(-4))^2+(y-0)^2=x-6#

#(x+4)^2+y^2=(x-6)^2#

#y^2=(x-6)^2-(x+4)^2#

#y^2=x^2-12x+36-x^2-8x-16#

#y^2=-20x+20=-20(x-1)#

graph{(y^2+20(x-1))(x-6)=0 [-14.23, 14.26, -7.11, 7.13]}