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Consider the parabola with a focus at the point (0, 7) and directrix y = 1. Which two equations can be used to correctly relate the distances from the focus and the directrix to any point (x, y) on the parabola?

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Explanation:

The equation that relates the distance from the focus and directrix to any point is


(y - k) {}^(2) = 4p(x - h)

or


(x - h) {}^(2) = 4p(y - k)

Since the directrix is a horizontal line, we will use the second equation.

The vertex lies halfway between the focus and directrix.

Since y=1, is perpendicular to the axis of symmetry, we are going to use the point (0,1) to represent the directrix.

Next, using the y values the number that lies between 7 and 1 is 4 so our vertex is


(0,4)

Our h is 0 and. k is 4.


(x - 0) {}^(2) = 4p(y - 4)


{x}^(2) = 4p(y - 4)

To find p, the equation of the focus is


(h,k + p)


(0,4 + p)


4 + p = 7


p = 3

So we have


{x}^(2) = 4(3)(y - 4)


{x}^(2) = 12(y - 4)

or


\frac{ {x}^(2) }{12} + 4 = y

User JdMR
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