How do you find an equation of the parabola with focus (2,2) and directrix x=-2?

1 Answer
May 17, 2017

#(y-2)^2=16(x-2)#

Explanation:

The key to solving this problem is knowing all your relevant equations. The formula for a parabola in vertex form is:

#(y-k)^2=4p(x-h)#
Vertex: #(h, k)#
Focus: #(p+h, k)#
Directrix: #x=-p+h#

Notice we chose the version that squares the #y#-value. This is because the given directrix is #x=-2#. If the directrix had been #y=-2#, then the formulas are different.

You are given the focus of #(2,2)#
#(h, k)=(2, 2)#
So #h=2# and #k=2#

You are also given the directrix #x=-2#
#x=-p+h=-2#

We already determined that #h=2# from above, so
#-p+2=-2#
#-p=-4# (subtract 2 from both sides)
#p=4# (divide both sides by -1)

Plugging these values back into the original equation gives
#(y-2)^2=4(4)(x-2)#
#(y-2)^2=16(x-2)#