Write the standard form of a parabola whose focus is (-3,0) and directrix x=3

Answer 1

Hello Tieaja,

Since the directrix is a vertical line, the parabola will be horizontal, so the equation will have the form:

(y-k)²=±4p*(x-h)

Also, if you draw x = 3 and label the focus (-3,0) on a coordinate plane, you'll notice that the focus is behind the directrix. The parabola equation will then be: (y-k)²=-4p*(x-h); p is the distance between the focus and the vertex; it also represents half the distance between the focus and the directrix.

Drawing the data on a coordinate plane will also show you that the distance between the focus and the directrix is 6 units, so p =3

To find the vertex coordinates (h,k), remember this: the vertex is halfway between the directrix and focus. So if it takes 6 units to get from directrix to focus, it will take 3 units to get from directrix to vertex. Thus h = 3 - 3 = 0. Since it is a horizontal parabola, the vertex and focus will have the same y-coordinate, therefore k = 0. so (h,k) = (0,0).

You can now plug p =3, (h,k) = (0,0) into (y-k)²=-4p*(x-h) and you get:

(y-0)²=-4(3)*(x-0)---> (y-0)²=-12*(x-0) ----> y²=-12x

Cheers.