Focus (2,-3), directrix x=5

Question

What is the equation for this problem?

Answer 1

4p(x-h) = (y-k)² is the equation of this parabola, which must be of the sideways-opening kind since the directrix is a vertical line.  The vertex is at (h,k).  It's NOT a function.  It should be helpful if you would sketch this. You'll see that the opening of this parabola will be leftward, so we must consider p to be negative.

 

k = -3. The distance from focus to vertex, and from vertex to directrix, is  |p|.  Consequently the distance from directrix to focus is 2p, and it's also the difference from 5 to 2, in the x direction. That is, 2|p| = 3.

 

With this, you can calculate h = 2+3/2 = 7/2.

 

Now you can substitute p, h & k into the equation.