Question

What is the equation of a parabola with a vertex at (-2, 0) and a directrix x = -4.

Answer 1

The distance between the vertex and the directrix is same as the distance from the vertex to the focus.
Distance from vertex to directrix = -2 - (-4) = -2 + 4 = 2
To find the focus, add 2 to the vertex: Focus is at (-2+2, 0) or (0,0)

A parabola has the property that all points on the parabola are equal distance from the focus and the directrix.

Let (x,y) be a general point on the parabola. The square of its distance from the focus is:
(x - 0)^2 + (y - 0)^2 = x^2 + y^2 --- (1)
The distance of (x,y) from the directrix x = -4 is: (x - (-4)) = x + 4.
Square this distance and equate it to (1)
x^2 + y^2 = (x + 4)^2
expand the right and simplify.

Answer 2

x^2 + y^2 = (x + 4)^2
x^2 + y^2 = x^2 + 8x + 16
y^2 = 8x + 16
y^2 - 16 = 8x
x = 1/8y^2 - 2

This is a horizontal parabola that opens to the right and its equation is x = 1/8y^2 - 2

Answer 3

Thank you so much you helped a lot!

Answer 4

You are welcome.