Find the Parabola with Focus (3,5) and Directrix y=1 (3,5) , y=1

$(3, 5)$ , $y = 1$

Since the directrix is vertical, use the equation of a parabola that opens up or down.

${(x - h)}^{2} = 4 p (y - k)$

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.

$(3, \frac{5 + 1}{2})$

Simplify the vertex.

Add $5$ and $1$ .

$(3, \frac{6}{2})$

Divide $6$ by $2$ .

$(3, 3)$

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 $p$ .

$p = 5 - 3$

Subtract $3$ from $5$ .

$p = 2$

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

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

Simplify.

${(x - 3)}^{2} = 8 (y - 3)$

Find the Parabola with Focus (3,5) and Directrix y=1 (3,5) , y=1

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