Find the Parabola with Focus F(0,8) and Directrix y=-8 (0,8) ; y=-8

$(0, 8)$ ; $y = - 8$

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.

$(0, \frac{8 - 8}{2})$

Simplify the vertex.

Cancel the common factor of $8 - 8$ and $2$ .

Factor $2$ out of $8$ .

$(0, \frac{2 \cdot 4 - 8}{2})$

Factor $2$ out of $- 8$ .

$(0, \frac{2 \cdot 4 + 2 \cdot - 4}{2})$

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

$(0, \frac{2 \cdot (4 - 4)}{2})$

Cancel the common factors.

Factor $2$ out of $2$ .

$(0, \frac{2 \cdot (4 - 4)}{2 (1)})$

Cancel the common factor.

$(0, \frac{2 \cdot (4 - 4)}{2 \cdot 1})$

Rewrite the expression.

$(0, \frac{4 - 4}{1})$

Divide $4 - 4$ by $1$ .

$(0, 4 - 4)$

Subtract $4$ from $4$ .

$(0, 0)$

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 $p$ .

$p = 8 - 0$

Subtract $0$ from $8$ .

$p = 8$

Step 4

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

${(x - 0)}^{2} = 4 (8) (y - 0)$

Step 5

Simplify.

$x^{2} = 32 y$

Step 6

Find the Parabola with Focus F(0,8) and Directrix y=-8 (0,8) ; y=-8

Related Questions

Support LCMGCF.com by disabling your adblocker.