Find the Parabola with Focus (-7,5) and Directrix y=-11 (-7,5) y=-11

$(- 7, 5)$ $y = - 11$

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

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

Find the vertex.

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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.

$(- 7, \frac{5 - 11}{2})$

Simplify the vertex.

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Subtract $11$ from $5$ .

$(- 7, \frac{- 6}{2})$

Divide $- 6$ by $2$ .

$(- 7, - 3)$

Find the distance from the focus to the vertex.

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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 $p$ .

$p = 5 + 3$

Add $5$ and $3$ .

$p = 8$

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

${(x + 7)}^{2} = 4 (8) (y + 3)$

Simplify.

${(x + 7)}^{2} = 32 (y + 3)$

Find the Parabola with Focus (-7,5) and Directrix y=-11 (-7,5) y=-11

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