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Question 166083: What is the equation for the parabola with focus (-2,-2) and directrix y=4?
Found 2 solutions by midwood_trail, gonzo:
Answer by midwood_trail(310)   (Show Source):
You can put this solution on YOUR website! What is the equation for the parabola with focus (-2,-2) and directrix y = 4?
We are dealing with the equation: y = ax^2
We need to find a in the equation y = ax^2
We use this fact:
Directrix: y= -1/4a
We are given the directrix to be 4.
So, plug and chug.
4 = -1/4a
Divide both sides by a.
4a = -1/4
Divide both sides by 4 to find a.
a = -1/16
The equation is y = (-1/16)x^2
Answer by gonzo(654) (Show Source):
You can put this solution on YOUR website! my solution to this problem is as follows:
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D (directrix) = (x,4)
F (Focus) = (-2,-2)
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The Vertex (V) is in between the Focus (F) and the directrix (D).
if D is at (x,4), and F is at (-2,-2), then V is half the distance between F and D which would be half the difference in the y values which would be 4 - (-2) = 6 / 2 = 3.
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p = the distance between the vertex (V and the directrix (D).
p = 3.
Vertex is between Focus (F) and directrix (D) in a vertical direction, so it has the same y coordinate as F, but a different x coordinate which is between D and F.
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what you have now is the following:
D at (x,4) - x can be any value, but for analysis of distances between V and F, you can assume it is -2 to put it in line with them.
V at (-2,1)
F at (-2,-2)
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the general equation for the parabola is (y-k) = [1/(4*p)] * (x-h)^2
the vertex of the parabola is at (h,k)
since the vertex is at (-2,1), then h - -2, and k = 1
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substituting into the equation, we get:
(y-1) = (1/4)*p * (x+2)^2
we calculated p to be the distance between V and D which was 3, so substituting 3 for p, we get:
(y-1) = [1/(4*3)] * (x+2)^2
simplifying, we get:
(y-1) = (1/12) * (x+2)^2
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the direction of the parabola is determined by where the focus is in relation to the vertex and the directrix.
since the directrix is on top, then comes the vertex further down then comes the focus on the bottom, this parabola opens downward.
for that reason, the 1/12 has to be negative.
the equation becomes:
(y-1) = -(1/12)*(x+2)^2
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we can add 1 to each side of the equation to get:
y = (-1/12) * (x+2)^2 + 1
which is the same equation, only now it can be plotted easily, which you can see right below:
look below the graph for further comments.
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in order to prove this equation is correct, you need to plot some points and show that the distance from the focal point to any point on the graph is the same distance from the point on the graph to the directrix.
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for example:
take the vertex (-2,1)
that's a point on the graph.
the corresponding point on the directrix directly above it is (-2,4)
the distance from the vertex to the directrix is sqrt (0^2 + 3^2) = sqrt(9) = 3.
the focus is at (-2,-2).
the distance from the vertex to the focus is sqrt(0^2 + 3^2) = sqrt(9) = 3.
since the distance from the vertex to the directrix is the same as the distance to the focus, the equation of the graph looks good (so far).
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take any other point on the graph and see if this relationship holds.
take x = 8
if x = 8, then y = ?????
y = [(-1/12) * (x+2)^2] + 1
y = [(-1/12) * (8+2)^2] + 1
y = [(-1/12) * (10)^2]+ 1
y = [(-1/12) * 100] + 1
y = -7.3333333
point on the graph is (8,-7.33333)
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distance from the point on the graph to the focus point is the distance between (-2,-2) and (8,-7.33333)
this distance is:
sqrt ( [-2-8]^2 + [-2-(-7.3333)]^2
= sqrt (-10^2 + 5.3333^2)
= sqrt (128.44444)
= 11.33333
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distance from the point on the graph to the directrix is the distance between (8,4) and (8,-7.33333) = sqrt (0^2 + 11.33333^2) = sqrt (11.33333^2) = 11.33333
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the distance between the point on the graph and the focus and the point on the graph and the directrix is the same.
the equation looks good.
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an additional point:
the parabola is a quadratic equation.
it can be in the form of ax^2 + bx + c
it can also be in the form of a (x-h) + k
the form of a (x-h) + k is closely related to the equation shown above for the parabola which is (1/4p) * (x-h) + k
the relationship between a and (1/4)p is that a = (1/4p) and p = (1/4a)
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a good website to explain in simple terms (relatively) can be found at:
http://home.alltel.net/okrebs/page64.html
good luck,
this is not all that easy to understand. it takes a few goings through to understand it.
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