Find the Parabola with Focus (4,0) and Directrix y=10 (4,0) y=10

$(4, 0)$ $y = 10$

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.

$(4, \frac{0 + 10}{2})$

Simplify the vertex.

Cancel the common factor of $0 + 10$ and $2$ .

Factor $2$ out of $0$ .

$(4, \frac{2 \cdot 0 + 10}{2})$

Factor $2$ out of $10$ .

$(4, \frac{2 \cdot 0 + 2 \cdot 5}{2})$

Factor $2$ out of $2 \cdot 0 + 2 \cdot 5$ .

$(4, \frac{2 \cdot (0 + 5)}{2})$

Cancel the common factors.

Factor $2$ out of $2$ .

$(4, \frac{2 \cdot (0 + 5)}{2 (1)})$

Cancel the common factor.

$(4, \frac{2 \cdot (0 + 5)}{2 \cdot 1})$

Rewrite the expression.

$(4, \frac{0 + 5}{1})$

Divide $0 + 5$ by $1$ .

$(4, 0 + 5)$

Add $0$ and $5$ .

$(4, 5)$

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 $p$ .

$p = 0 - 5$

Subtract $5$ from $0$ .

$p = - 5$

Step 4

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

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

Step 5

Simplify.

${(x - 4)}^{2} = - 20 (y - 5)$

Step 6

Find the Parabola with Focus (4,0) and Directrix y=10 (4,0) y=10

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