derive the equation of the parabola with a focus at (2,4) and a directrix of y=8

Answer 1

Given: Focus at (2,4) and directrix at y = 8

Find: Equation of parabola.

SOLUTION:

On a piece of paper, sketch the focus (2,4) and the directrix y=8.

The focus is below the directrix.

So, the parabola opens downward.

The general form of the equation is: -(4p)y = x^2

The vertex is equal-distance between the focus and directrix.

In other words, the vertex is exactly in the middle of the focus and directrix.

Therefore, the vertex is at (h,k) = (2, 6)

The distance from the vertex to the focus is 2.

Also, the distance from the vertex to the directrix is 2.

So, p = 2.

The general form of the equation, with vertex at (h,k) is:

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

The equation of this parabola is:

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

-(4*2)(y - 6) = (x - 2)^2

-8(y - 6) = (x - 2 )^2

If you don't like that format, we can change it!

(y - 6) = (-1/8)(x - 2 )^2

y = (-1/8)(x - 2)^2 + 6