Question

What is the vertex form of the equation of the parabola with a focus at (8,-5) and a directrix of `y=-6`?

Answer 1

The directrix is a horizontal line, therefore, the vertex form is:
`y=a(x-h)^2+k" [1]"`
`a = 1/(4f)" [2]"`
The focus is `(h,k+f)" [3]"`
The equation of the directrix is `y=k-f" [4]"`

Explanation:

Given that the focus is `(8,-5)`, we can use point [3] to write the following equations:

`h = 8" [5]"`
`k + f = -5" [6]"`

Given that the equation of the directrix is `y = -6`, we can use equation [4] to write the following equation:

`k - f = -6" [7]"`

We can use equations [6] and [7] to find the values of k and f:

`2k = -11`
`k = -11/2`

`-11/2 + f = -5 = -10/2`
`f = 1/2`

Use equation [2] to find the value of "a":

`a = 1/(4f)`
`a = 1/(4(1/2)`
`a = 1/2`

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

`y = 1/2(x - 8)^2 -11/2" [8]"`

Equation [8] is the desired equation.