Question

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?

Answer 1

Step-by-step 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

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