Find the Parabola with Focus (-3,-5) and Directrix x=-7 (-3,-5) x=-7

$(- 3, - 5)$ $x = - 7$

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

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

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.

$(\frac{- 3 - 7}{2}, - 5)$

Simplify the vertex.

Subtract $7$ from $- 3$ .

$(\frac{- 10}{2}, - 5)$

Divide $- 10$ by $2$ .

$(- 5, - 5)$

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 $p$ .

$p = - 3 + 5$

Add $- 3$ and $5$ .

$p = 2$

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

${(y + 5)}^{2} = 4 (2) (x + 5)$

Simplify.

${(y + 5)}^{2} = 8 (x + 5)$

Find the Parabola with Focus (-3,-5) and Directrix x=-7 (-3,-5) x=-7

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