Find the Parabola with Focus (5,6) and Directrix x=-2 focus (5,6) ; directrix x=-2

focus $(5, 6)$ ; directrix $x = - 2$

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.

$(\frac{5 - 2}{2}, 6)$

Subtract $2$ from $5$ .

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

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

$p = 5 - \frac{3}{2}$

Simplify.

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$p = 5 \cdot \frac{2}{2} - \frac{3}{2}$

Combine $5$ and $\frac{2}{2}$ .

$p = \frac{5 \cdot 2}{2} - \frac{3}{2}$

Combine the numerators over the common denominator.

$p = \frac{5 \cdot 2 - 3}{2}$

Simplify the numerator.

Multiply $5$ by $2$ .

$p = \frac{10 - 3}{2}$

Subtract $3$ from $10$ .

$p = \frac{7}{2}$

Step 4

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

${(y - 6)}^{2} = 4 (\frac{7}{2}) (x - \frac{3}{2})$

Step 5

Simplify.

${(y - 6)}^{2} = 14 (x - \frac{3}{2})$

Step 6

Find the Parabola with Focus (5,6) and Directrix x=-2 focus (5,6) ; directrix x=-2

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