Find the Parabola with Focus (0,7) and Directrix y=3 focus (0,7) directrix y=3

focus $(0, 7)$ directrix $y = 3$

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.

$(0, \frac{7 + 3}{2})$

Simplify the vertex.

Add $7$ and $3$ .

$(0, \frac{10}{2})$

Divide $10$ by $2$ .

$(0, 5)$

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 $p$ .

$p = 7 - 5$

Subtract $5$ from $7$ .

$p = 2$

Step 4

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

${(x - 0)}^{2} = 4 (2) (y - 5)$

Step 5

Simplify.

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

Step 6

Find the Parabola with Focus (0,7) and Directrix y=3 focus (0,7) directrix y=3

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