Question

A parabola has focus (5, 7) and directrix y=0 (the x-axis). Write an equation that says point (x, y) is

on the parabola. You do not need to put it into vertex form. If it's helpful, draw a sketch of the
parabola.

Answer 1

14(y -7/2) = (x -5)²

Explanation:

You want an equation for the parabola with focus (5, 7) and directrix y=0.

Equation

The vertex of a parabola is halfway between the focus and the directrix. When the directrix is a horizontal line, the y-coordinate of the vertex is the average of the y-coordinates of the focus and directrix. The x-coordinate of the vertex is the same as that of the focus.

The scale factor in the vertex form equation is 1/(4p), where p is the distance from the vertex to the directrix.

Application

Here, the vertex y-coordinate is (7+0)/2 = 3.5. The vertex x-coordinate is 5. The distance from the vertex to the directrix is 3.5 -0 = 3.5.

Then the vertex form equation for the parabola can be written as ...

y = a(x -h)² +k . . . . . . . . parabola with scale factor 'a' and vertex (h, k)

y = 1/(4·3.5)·(x -5)² +3.5 . . . . . parabola with scale factor 1/(4·3.5) and vertex (5, 3.5)

The fraction can be eliminated to give the form ...

14(y -3.5) = (x -5)²