Question

How do you find the equation of a parabola with vertex at the origin and directrix x=3?

Answer 1

`y^2+12x=0`

Explanation:

As we have vertex at the origin i.e. `(0,0)` and directrix is `x=3`, a line parallel to `y`-axis,

it must have a focus at `(-3,0)`.

Equation of the parabola represents locus of a point `(x,y)`, which moves so that its distance from `x=3` and `(-3,0)` are equal. Hence, equation of parabola is

`(x-(-3))^2+(y-0)^2=(x-3)^2`

or `(x+3)^2+y^2=(x-3)^2`

or `x^2+6x+9+y^2=x^2-6x+9`

or `y^2+12x=0`
graph{y^2+12x=0 [-27.59, 12.41, -10.08, 9.92]}