Question

A parabola can be drawn given a focus of (−5,3) and a directrix of

y=1. Write the equation of the parabola in any form.

Answer 1

Explanation:

y = 1 is a horizontal line

Vertex is halfway between the directrix and focus on a vertical line through x = (-5, 3).

(1 + 3)/2 = (-5, 2)

The shortest distance from the directrix to the focus is 3 - 1 = 2 units

so points (-5 - 2, 3) = (-7, 3) and (-5 + 2, 3) = (-3, 3) are on the parabola.

The vertex form of a parabola is

y = a(x - h)² + k

y = a(x - (-5))² + 2

y = a(x + 5)² + 2

plugging in one of the other known points

3 = a(-7 + 5)² + 2 or 3 = a(-3 + 5)² + 2

1 = a(-2)² 1 = a(2)²

a = 1/4 a = 1/4

y = ¼(x + 5)² + 2

to find the standard form, expand the vertex form

y = ¼(x² + 10x + 25) + 2

y = 0.25x² + 2.5x + 6.25 + 2

y = 0.25x² + 2.5x + 8.25

or

4y = x² + 10x + 33