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

$(2, 2)$ $x = 8$

Step 1

Since the directrix is horizontal, use the equation of a parabola that opens left or right.

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

Step 2

Find the vertex.

The vertex $(h, k)$ is halfway between the directrix and focus. Find the $x$ coordinate of the vertex using the formula $x = \frac{x coordinate of focus + directrix}{2}$ . The $y$ coordinate will be the same as the $y$ coordinate of the focus.

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

Simplify the vertex.

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

Factor $2$ out of $2$ .

$(\frac{2 \cdot 1 + 8}{2}, 2)$

Factor $2$ out of $8$ .

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

Factor $2$ out of $2 \cdot 1 + 2 \cdot 4$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $1 + 4$ by $1$ .

$(1 + 4, 2)$

Add $1$ and $4$ .

$(5, 2)$

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

$p = 2 - 5$

Subtract $5$ from $2$ .

$p = - 3$

Step 4

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

${(y - 2)}^{2} = 4 (- 3) (x - 5)$

Step 5

Simplify.

${(y - 2)}^{2} = - 12 (x - 5)$

Step 6

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

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