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Question 1081914: The focus of a parabola is (3,−7) and the directrix is y=−4.
What is an equation of the parabola?
(x−3)2=−6(y+5.5)
(x−3)2=−3(y+10)
(x−3)2=−12(y+10)
(x−3)2=−1.5(y+5.5)
Found 2 solutions by Boreal, KMST:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! The coordinates of the focus are (h, k + (1/4a)), where h=3, and k+(1/4a)=-7
k+(1/4a)=-7
The vertex is half way between the focus and the directrix, or at (3, -5.5), so k=-5.5
-5.5+(1/4a)=-7
(1/4a)=-1.5
1=-6a
a=(-1/6)
The equation is y=(-1/6)(x-3)^2-5.5
or (y+5.5)=(-1/6)(x-3)^2
or -6(y+5.5)=(x-3)^2
Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website! From the definition of parabola, you can find the answer,
without needing to resort to a memorized formula.
The definition says that a parabola is the set of all points
whose distances to vertex and d directrix are equal.
1) The vertex must be halfway between y=-4 and focus (3,-7),
so the vertex is (3,-5.5).
That point satisfies
 and  ,
but not the other two equations.
2) A point with y=-7 is at a distance
-4 - (-7) = -4 + 7 = 3 from the directrix,
so it must be at distance 3 from focus (3,-7).
That only happens for points (0,-7) and (6,-7).
Substituting x=0 and y=-7, you find that
 equals
 ,
so is the equation you are looking for.
Your teacher may have told you that the equation of a parabola
with vertex (h,k), directrix y=k-a and focus (h,k+a) is
 .
If so, and if you memorized that formula,
you easily see that in this case h=3.
You can find k=-5.5 as the average of
the -4 in y=-4, and the -7 in (3,-7),
and you know that k+a=-5.5+a
is the y-coordinate of the focus,
so -5.5+a = -7, meaning a= -7 +5.5 = -1.5.
Applying the memorized formula gives you the same result,
without needing to understand why you do what you do.
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