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

1 Answer
Aug 26, 2017

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