Find the Parabola with Focus (-5,5) and Directrix y=-1 (-5,5) y=-1

$(- 5, 5)$ $y = - 1$

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.

$(- 5, \frac{5 - 1}{2})$

Simplify the vertex.

Subtract $1$ from $5$ .

$(- 5, \frac{4}{2})$

Divide $4$ by $2$ .

$(- 5, 2)$

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 $p$ .

$p = 5 - 2$

Subtract $2$ from $5$ .

$p = 3$

Step 4

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

${(x + 5)}^{2} = 4 (3) (y - 2)$

Step 5

Simplify.

${(x + 5)}^{2} = 12 (y - 2)$

Step 6

Find the Parabola with Focus (-5,5) and Directrix y=-1 (-5,5) y=-1

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