Find the Parabola with Focus (7,-6) and Directrix x=5 (7,-6) x=5

$(7, - 6)$ $x = 5$

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.

$(\frac{7 + 5}{2}, - 6)$

Simplify the vertex.

Add $7$ and $5$ .

$(\frac{12}{2}, - 6)$

Divide $12$ by $2$ .

$(6, - 6)$

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 $p$ .

$p = 7 - 6$

Subtract $6$ from $7$ .

$p = 1$

Step 4

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

${(y + 6)}^{2} = 4 (1) (x - 6)$

Step 5

Simplify.

${(y + 6)}^{2} = 4 (x - 6)$

Step 6

Find the Parabola with Focus (7,-6) and Directrix x=5 (7,-6) x=5

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