Question

What is the standard form of the equation of the parabola with a directrix at `x=9` and a focus at `(8,4)`?

Answer 1

The standard form is: `x = -1/2y^2 +4y+1/2`

Explanation:

Because the directrix is a vertical line, one knows that the vertex form of the equation for the parabola is:

`x = 1/(4f)(y-k)^2+h" [1]"`

where `(h,k)` is the vertex and `f` is the signed horizontal distance from the vertex to the focus.

The x coordinate of the vertex halfway between the directrix and the focus :

`h = (9+8)/2`

`h = 17/2`

Substitute into equation [1]:

`x = 1/(4f)(y-k)^2+17/2" [2]"`

The y coordinate of the vertex is the same as the y coordinate of the focus:

`k = 4`

Substitute into equation [2]:

`x = 1/(4f)(y-4)^2+17/2" [3]"`

The value of `f` is the signed horizontal distance from the vertex to the focus#

`f = 8-17/2`

`f = -1/2`

Substitute into equation [3]:

`x = 1/(4(-1/2))(y-4)^2+17/2`

This is the vertex form:

`x = -1/2(y - 4)^2+17/2`

Expand the square:

`x = -1/2(y^2 -8y+16)+17/2`

Use the distributive property:

`x = -1/2y^2 +4y-8+17/2`

Combine like terms:

`x = -1/2y^2 +4y+1/2`

Here is a graph of the standard form, the focus, the vertex, and the directrix:

www.desmos.com/calculator