Find the Parabola with Focus (0,5) and Directrix x=-7 (0,5) x=-7

$(0, 5)$ $x = - 7$

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.

$(\frac{0 - 7}{2}, 5)$

Simplify the vertex.

Subtract $7$ from $0$ .

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

Move the negative in front of the fraction.

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

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 $p$ .

$p = 0 + \frac{7}{2}$

Add $0$ and $\frac{7}{2}$ .

$p = \frac{7}{2}$

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

${(y - 5)}^{2} = 4 (\frac{7}{2}) (x + \frac{7}{2})$

Simplify.

${(y - 5)}^{2} = 14 (x + \frac{7}{2})$

Find the Parabola with Focus (0,5) and Directrix x=-7 (0,5) x=-7

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