Question

What is the equation in standard form of the parabola with a focus at (18,24) and a directrix of y= 27?

Answer 1

`y = -1/6x^2+6x- 57/2 larr` standard form

Explanation:

We know that the standard form for the equation of a parabola with a horizontal directrix is

`y = ax^2+bx+c`

but, because we are given the focus and the equation of the directrix, it is easier to start with the corresponding vertex form

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

and then convert to standard form.

We know that the x coordinate, "h", of the vertex is the same as the x coordinate of the focus:

`h = 18`

Substitute into equation [1]:

`y = a(x-18)^2+k" [2]"`

We know that the y coordinate, "k", of the vertex is the midpoint between the focus and the directrix:

`k = (24+27)/2`

`k = 51/2`

Substitute into equation [2]:

`y = a(x-18)^2+51/2" [3]"`

The focal distance, "f", is the signed vertical distance from the vertex to the focus:

`f = 24-51/2`

`f = -3/2`

We know that `a = 1/(4f)`

`a = 1/(4(-3/2)`

`a = -1/6`

Substitute into equation [3]:

`y = -1/6(x-18)^2+51/2`

Expand the square:

`y = -1/6(x^2-36x+ 324)+51/2`

Distribute the `-1/6`:

`y = -1/6x^2+6x- 54+51/2`

Combine like terms:

`y = -1/6x^2+6x- 57/2 larr` standard form