Find the Parabola with Focus (4,3) and Directrix y=13 (4,3) y=13

$(4, 3)$ $y = 13$

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.

$(4, \frac{3 + 13}{2})$

Simplify the vertex.

Add $3$ and $13$ .

$(4, \frac{16}{2})$

Divide $16$ by $2$ .

$(4, 8)$

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 $p$ .

$p = 3 - 8$

Subtract $8$ from $3$ .

$p = - 5$

Step 4

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

${(x - 4)}^{2} = 4 (- 5) (y - 8)$

Step 5

Simplify.

${(x - 4)}^{2} = - 20 (y - 8)$

Step 6

Find the Parabola with Focus (4,3) and Directrix y=13 (4,3) y=13

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