Find the Parabola with Focus (8,-8) and Directrix y=-6 (8,-8) y=-6

$(8, - 8)$ $y = - 6$

Step 1

Since the directrix is vertical, use the equation of a parabola that opens up or down.

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

Step 2

Find the vertex.

The vertex $(h, k)$ is halfway between the directrix and focus. Find the $y$ coordinate of the vertex using the formula $y = \frac{y coordinate of focus + directrix}{2}$ . The $x$ coordinate will be the same as the $x$ coordinate of the focus.

$(8, \frac{- 8 - 6}{2})$

Simplify the vertex.

Cancel the common factor of $- 8 - 6$ and $2$ .

Factor $2$ out of $- 8$ .

$(8, \frac{2 \cdot - 4 - 6}{2})$

Factor $2$ out of $- 6$ .

$(8, \frac{2 \cdot - 4 + 2 \cdot - 3}{2})$

Factor $2$ out of $2 \cdot - 4 + 2 \cdot - 3$ .

$(8, \frac{2 \cdot (- 4 - 3)}{2})$

Cancel the common factors.

Factor $2$ out of $2$ .

$(8, \frac{2 \cdot (- 4 - 3)}{2 (1)})$

Cancel the common factor.

$(8, \frac{2 \cdot (- 4 - 3)}{2 \cdot 1})$

Rewrite the expression.

$(8, \frac{- 4 - 3}{1})$

Divide $- 4 - 3$ by $1$ .

$(8, - 4 - 3)$

Subtract $3$ from $- 4$ .

$(8, - 7)$

Step 3

Find the distance from the focus to the vertex.

The distance from the focus to the vertex and from the vertex to the directrix is $| p |$ . Subtract the $y$ coordinate of the vertex from the $y$ coordinate of the focus to find $p$ .

$p = - 8 + 7$

Add $- 8$ and $7$ .

$p = - 1$

Step 4

Substitute in the known values for the variables into the equation ${(x - h)}^{2} = 4 p (y - k)$ .

${(x - 8)}^{2} = 4 (- 1) (y + 7)$

Step 5

Simplify.

${(x - 8)}^{2} = - 4 (y + 7)$

Step 6

Find the Parabola with Focus (8,-8) and Directrix y=-6 (8,-8) y=-6

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