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

1 Answer
Aug 7, 2018

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

Explanation:

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]}