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Question 370136: What is the equation of the parabola with a focus at (-3,0) and a directrix x=3?
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! x = 3 is the equation of a vertical line. A parabola with a vertical line directrix will be a parabola that opens horizontally (to the right or to the left).
Since the focus, (-3, 0), is to the left of the directrix, this parabola will open to the left (so that the focus is inside the "bowl" of the parabola).
A parabols that opens to the left will have an equation of the form:
The squaring of y is what makes the parabola open horizontally,
The minus in front of 4p makes it open to the left. (A positive would makeit open to the right.)
The h and k in the equation are the coordinates of the vertex. The vertex is halfway between the focus and the directrix, The point which is halfway between (-3, 0) and the line x = 3 would be (0, 0). So the vertex is (0, 0) which makes h = k = 0.
The "p" in the equation is the distance between the focus and directrix. The distance from the focus, (-3, 0), and the vertex, (0, 0), is 3. So p = 3.
Replacing the h, k and p into
we get:
which simplifies as follows:
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