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Question 173639: find the equation of the parabola that satisfies the given conditions.
Focus: (7,0), directrix: x=1
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! focus = (7,0)
directrix = x = 1
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looks like this is a parabolic that is symmetric to the x axis (head points to right or left rather than up or down).
that's because the directrix line is parallel to the y axis.
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if the focus is at (7,0) and one point on the directrix is (1,0), then the distance between the directrix and the focus is 7-1 = 6.
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the vertex is right in the middle between the focus and the directrix, so the vertex must be a distance of 3 from the directrix and a distance of 3 from the focus.
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that puts the vertex at (4,0).
that makes it a distance of 3 from (1,0) and a distance of 3 from (7,0).
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you have the required information to generate the equation.
focus = (7,0)
vertex = (4,0)
directrix = (1,y)
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since the directrix is to the left of the vertex, that means the head of this parabola will be point to the left and the tails will be pointing to the right.
that happens when a is positive.
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general form of the equation of a parabola in the up/down direction is:
general form of the equation of a parabola in the right/left direction is:
where:
a = 1/4d
where:
d = distance between focus and vertex.
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if you have d, you can derive a by the formula:
a = 1/4d
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if you have a, you can derive d by the formula:
d = 1/4a
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since the vertex of a parabola is (h,k), we have:
h = 4
k = 0
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since the distance from the focus to the vertex is 3, we have d = 3 we can derive a by using the formula:
a = 1/4d = 1/(4)*(3) = 1/12
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the general form of the equation of a parabola in the right / left direction is:
becomes:
this looks like:
look below the graph for further comments.
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not shown on this graph is the directrix line which is a vertical line at x = 1.
also not shown on this graph is the focus which is at (7,0).
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to graph the equation for a left/right facing parabola, i took the equation:
and solved for y, so that the variable y is on the left and everything else is on the right.
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a fairly decent treatment of focus and directrix of parabola can be found at the following website:
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_conics_directrix.xml
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this is treating a parabola in he up/down direction.
your case was different but the same concepts apply as long as you keep in mind that you replace x with y, and you replace h with k, and k with h to convert the parabola to left/right direction:
up/down direction:
y = a*(x-h)^2 + k
left/right direction:
x = a*(y-k)^2 + h
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