Question

Write an equation for a parabola in which the set of all points in the plane are equidistant from the focus and line. F(–2, 0); x = 2

Answer 1

The line is called the directrix.  Here we have a vertical directrix, so a parabola sideways from usual.

Geometry is best done with squared distances.  The squared distance from an arbitrary point (x,y) to the vertical line x=2 is $(x-2)^2.$

We equate that to the squared distance of (x,y) to the focus (-2,0): 

$(x-2)^2 = (x - -2)^2 + (y - 0)^2$

$x^2 -4x + 4=x^2 +4x +4 + y^2$

$-8x = y^2$

We could call that done.  A more standard form might be

$x =- \dfrac 1 8 \ y^2$