Question

Write the equation for a parabola with a focus at (2, 2) and a directrix at x = 8.

Answer 1

Let  P(x,y) be a point on the parabola.  The definition of a parabola is the squared distance from P to focus F(2,2) equals the squared distance to the directrix x=8.

squared distance from P(x,y) to focus F(2,2)

(x - 2)² + (y - 2)²

squared distance from P(x,y) to the directrix x=8.

(x-8)²

Those are equal in a parabola,

(x - 2)² + (y - 2)² = (x-8)²

x² - 4x + 4 + y² - 4y + 4 = x² - 16x + 64

y² - 4y - 46 = -12x

That's a sideways parabola,

x = -(y² - 4y - 46)/12