What is the standard form of the equation of the parabola with a directrix at x=-3 and a focus at (1,-1)?

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Answer 1 · 2018-08-07T08:47:31

$x=1/8(y+1)^2-8$

Parabola is the locus of a point which moves so that its distance from a given point called focus and a given line called directrix is always equal.

Let the point be $(x,y)$ . Its distance from focus $(1,-1)$ is

$sqrt((x-1)^2+(y+1)^2)$

and its distance from directrix $x=-3$ or $x+3=0$ is $x+3$

Hence equation of parabola is $sqrt((x-1)^2+(y+1)^2)=x+3$

and squaring $(x-1)^2+(y+1)^2=(x+3)^2$

i.e. $x^2-2x+1+y^2+2y+1=x^2+6x+9$

i.e. $y^2+2y-7=8x$

or $8x=(y+1)^2-8$

or $x=1/8(y+1)^2-8$

graph{(y^2+2y-7-8x)((x-1)^2+(y+1)^2-0.01)(x+3)=0 [-11.17, 8.83, -5.64, 4.36]}