To determine the equation of the parable whose string minimum joining the focal points A(3,5) and B(3,-3)
Are you trying to find the equation of two parabolas:
Parabola 1 has focal point A(3,5).
Parabola 2 has focal point B(3,-3).
But I don't understand the 'string minimum' condition you talk about. Are you minimising some sort of distance between these two parabolas and their focal points?
Alright, hold on a second. Mr. Fantastic corrected me on some little thing, so I want to feel really good that I got this right.
OK, I started drawing a diagram of the two points. Then, I drew the line segment between them. Then, I found the midpoint. If I cheat ahead an look at the answer, I find that the midpoint between the two points is the focus of both parabolas. Also, the two points given lie on both parabolas.
Now, I am going to conjecture here a bit. I think that what you need to find are the possible equations of parabolas such that the focus is the minimum distance from both points. If you do this and follow the steps laid out in the other parabola help topic, I think you should come up with the right answer.