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Question 587098: Find the equation of the parabola with vertex at (3,-2), focus at (3,4).
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website! Find the equation of the parabola with vertex at (3,-2), focus at (3,4).
Let's plot the vertex and focus
The equation of a parabola with focus above or below the vertex is
(x - h)² = 4p(y - k)
where the vertex is (h,k) and p = distance from vertex to to focus,
taken as a positive number if the focus is above the vertex and
negative if the focus is below the vertex.
In this case the vertex = (h,k) = (3,-2)
By counting the units from the vertex up to the focus we see
that p = 6, and is taken positive since the focus is above the
vertex, which also means that the parabola opens upward,
So the equation is:
(x - h)² = 4p(y - k)
(x - 3)² = 4(6)(y - (-2))
(x - 3)² = 24(y + 2)
That's all you were asked to find. But you might have
been asked to graph it and find the directrix.
We draw the focal chord (also called "latus rectum")
which has length 4p = 4(6) = 24 and whose midpoint is
the focus:
And we can sketch in the parabola:
We can also find the directrix which is a horizontal line p = 6
units below the vertex, the dotted line below:
The directrix has equation y = -8
Edwin
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