What is the standard form of the equation of the parabola with a directrix at x=-5 and a focus at (-7,-5)?

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Answer 1 · 2017-10-31T15:25:01

The equation of the parabola is $(y+5)^2=-4x-24=-4(x+6)$

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

Therefore,

$x-(-5)=sqrt((x-(-7))^2+(y-(-5))^2)$

$x+5=sqrt((x+7)^2+(y+5)^2)$

Squaring and developing the $(x+7)^2$ term and the LHS

$(x+5)^2=(x+7)^2+(y+5)^2$

$x^2+10x+25=x^2+14x+49+(y+5)^2$

$(y+5)^2=-4x-24=-4(x+6)$

The equation of the parabola is $(y+5)^2=-4x-24=-4(x+6)$

graph{((y+5)^2+4x+24)((x+7)^2+(y+5)^2-0.03)(y-100(x+5))=0 [-17.68, 4.83, -9.325, 1.925]}