Question

How do you find the equation of a parabola giving the focus and directix

Answer 1

Explanation:

There are a few things to remember for this:

• The vertex is halfway between the focus and the directrix
• The parabola opens in the direction toward the focus and away from the directrix
• A useful formula is y = (1/(4p))(x -h)^2 +k, where p is the distance from the focus to the vertex, and (h, k) is the vertex). If the focus is above the directrix, p will be positive; negative otherwise.

You may see more complicated formulas that have you plug in focus and directrix values. I find it easier to remember the above, since it involves the vertex form you've already seen: y = a(x -h)^2 +k. The "trick" is that the scale factor (a) is 1/(4p).

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If you're graphing the result, as a check on your work, you can make sure that the parabola matches its geometric definition. The distance from any point on the parabola to the focus is the same as the distance from that point to the directrix. This is why the vertex is halfway between the focus and directrix: so the distances are the same.

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Example:

Focus (1, -3); directrix y=5.

Distance from focus to directrix: -3-5 = -8, so p = -4. The vertex is (1, 1), and the equation is ...

  y = -1/16(x -1)^2 +1