Find the Parabola with Focus (-7,-7) and Directrix x=-9 (-7,-7) , x=-9

$(- 7, - 7)$ , $x = - 9$

Step 1

Since the directrix is horizontal, use the equation of a parabola that opens left or right.

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

Step 2

Find the vertex.

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

$(\frac{- 7 - 9}{2}, - 7)$

Simplify the vertex.

Subtract $9$ from $- 7$ .

$(\frac{- 16}{2}, - 7)$

Divide $- 16$ by $2$ .

$(- 8, - 7)$

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 $x$ coordinate of the vertex from the $x$ coordinate of the focus to find $p$ .

$p = - 7 + 8$

Add $- 7$ and $8$ .

$p = 1$

Step 4

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

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

Step 5

Simplify.

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

Step 6

Find the Parabola with Focus (-7,-7) and Directrix x=-9 (-7,-7) , x=-9

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