Write an equation of a parabola in standard form using a focus and directrix.

Question

A parabola has a focus of (3,6) and the directrix is y=2. What is the equation for the parabola in standard form?

Answer 1

Since the directrix is y = 2 (a horizontal line), the axis is vertical.

 

The vertex and focus lie on the axis.  The vertex, (3, 4), is halfway between the directrix and focus.  The focus lies inside the arc of the parabola.  So, the parabola opens upward.

 

Equation of parabola has the form (x-3)² = 4p(y-6).

 

   p = distance from vertex to focus = 2

 

Equation of parabola:  (x-3)² = 8(y-6)

 

      x² - 6x + 9 = 8y - 48

 

                      8y = x² - 6x + 57

 

                        y = (1/8)x² - (3/4)x + (57/8)