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

1 Answer
Jan 21, 2017

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.