Question

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

Answer 1

`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]}