Question

Find the standard form of the equation of the parabola with a focus at (-2, 0) and a directrix at x = 2.

answers:

a) y2 = 4x

b)8y = x2

c)x = 1 divided by 8y2

d) y = 1 divided by 8x2

Answer 1

`y^2=(1)/(8)x`

Explanation:

The focus lies on the x axis and the directrix is a vertical line through x = 2. The parabola, by nature, wraps around the focus, or "forms" its shape about the focus. That means that this is a "sideways" parabola, a "y^2" type instead of an "x^2" type. The standard form for this type is

`(x-h)=4p(y-k)^2`

where h and k are the coordinates of the vertex and p is the distance from the vertex to either the focus or the directrix (that distance is the same; we only need to find one). That means that the vertex has to be equidistant from the focus and the directrix. If the focus is at x = -2 y = 0 and the directrix is at x = 2, midway between them is the origin (0, 0). So h = 0 and k = 0. p is the number of units from the vertex to the focus (or directrix). That means that p=2. We fill in our equation now with the info we have:

`(x-0)=4(2)(y-0)^2`

Simplify that a bit:

`x=8y^2`

Solving for y^2:

`y^2=(1)/(8)x`

Answer 2

Answer: x = -1/8y^2

Step-by-step explanation

Focus: (-2,0)

Directrix: x=2

It meets the criteria.