Find the Parabola with Focus (2,4) and Directrix y=8 (2,4) , y=8

$(2, 4)$ , $y = 8$

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.

$(2, \frac{4 + 8}{2})$

Simplify the vertex.

Cancel the common factor of $4 + 8$ and $2$ .

Factor $2$ out of $4$ .

$(2, \frac{2 \cdot 2 + 8}{2})$

Factor $2$ out of $8$ .

$(2, \frac{2 \cdot 2 + 2 \cdot 4}{2})$

Factor $2$ out of $2 \cdot 2 + 2 \cdot 4$ .

$(2, \frac{2 \cdot (2 + 4)}{2})$

Cancel the common factors.

Factor $2$ out of $2$ .

$(2, \frac{2 \cdot (2 + 4)}{2 (1)})$

Cancel the common factor.

$(2, \frac{2 \cdot (2 + 4)}{2 \cdot 1})$

Rewrite the expression.

$(2, \frac{2 + 4}{1})$

Divide $2 + 4$ by $1$ .

$(2, 2 + 4)$

Add $2$ and $4$ .

$(2, 6)$

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 $y$ coordinate of the vertex from the $y$ coordinate of the focus to find $p$ .

$p = 4 - 6$

Subtract $6$ from $4$ .

$p = - 2$

Step 4

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

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

Step 5

Simplify.

${(x - 2)}^{2} = - 8 (y - 6)$

Step 6

Find the Parabola with Focus (2,4) and Directrix y=8 (2,4) , y=8

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