Question

What is the equation of the parabola with a focus at (3,6) and a directrix of y= 0?

Answer 1

The vertex form of the equation for the parabola is:

`y = 1/12(x-3)^2+3`

Explanation:

The directrix is a horizontal line, therefore, the vertex form of the equation of the parabola is:

`y = a(x-h)^2+k" [1]"`

The x coordinate of the vertex, h, is the same as the x coordinate of the focus:

`h = 3`

The y coordinate of the vertex, k, is midpoint between the directrix and the focus:

`k = (6 + 0)/2 = 3`

The signed vertical distance, f, from the vertex to the focus is, also, 3:

`f = 6-3 = 3`

Find the value of "a" using the formula:

`a = 1/(4f)`

`a = 1/(4(3))`

`a = 1/12`

Substitute the values of h, k, and a into equation [1]:

`y = 1/12(x-3)^2+3" [2]"`