What is the equation of the parabola with a focus at (-5,-8) and a directrix of y= -3?

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Answer 1 · 2017-12-13T14:51:08

#y=-1/10x^2-x-8#

Parabola is the path traced by a point so that it's distance from a given point called focus and a given line called directrix is always equal.

Let the point on parabola be #(x,y)#.

It's distance from focus #(-5,-8)# is #sqrt((x+5)^2+(y+8)^2)# and it's distance from line #y=-3# or #y+3=0# is #|y+3|#.

Hence the equation of the parabola with a focus at #(-5,-8)# and a directrix of #y= -3?# is

#sqrt((x+5)^2+(y+8)^2)=|y+3|#

or #(x+5)^2+(y+8)^2)=(y+3)^2#

or #x^2+10x+25+y^2+16y+64=y^2+6y+9#

or #10y=-x^2-10x-80#

or #y=-1/10x^2-x-8#

graph{ (10y+x^2+10x+80)(y+3)((x+5)^2+(y+8)^2-0.1)=0 [-15, 5, -10, 0]}