equation for a parabola

label Algebra
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write an equation for the parabola whose focus is (5,-2) and whose directrix is y=6

Dec 11th, 2014

The focus is a point and the directrix is a line.

The parabola has the equation

     (x - h)² = 4p(y - k)

where (h,k) is the vertex, p is the distance from
the vertex to the focus and also the distance
from the vertex to the focus. 

Lets draw the focus F(3,2). I'll put an "o" there.
And I'll draw the directrix y=6 which is a horizontal
line 6 units above the x-axis.  I'll draw it green



The vertex is halfway between the focus and the
directrix.  So the vertex is at the point V(3,4).

I'll plot the vertex with an o.



That means that the vertex (h,k) = (3,4)

So in the equation 

     (x - h)² = 4p(y - k)

we can now put in h and k:

     (x - 3)² = 4p(y - 4)

Now we only need p, which is the
distance from the vertex to the focus 
and also the distance from the
vertex to the directrix. 

So the distance from V(3,4) to F(3,2)
is 2 units, so p = 2.

In the equation

     (x - 3)² = 4p(y - 4)     

we can put in 2 for p and have

     (x - 3)² = 4(2)(y - 4)

or

     (x - 3)² = 8(y - 4)


Dec 11th, 2014

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