Find the Parabola with Focus (4,-3) and Directrix y=-6 (4,-3) y=-6

$(4, - 3)$ $y = - 6$

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.

$(4, \frac{- 3 - 6}{2})$

Simplify the vertex.

Subtract $6$ from $- 3$ .

$(4, \frac{- 9}{2})$

Move the negative in front of the fraction.

$(4, - \frac{9}{2})$

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

$p = - 3 + \frac{9}{2}$

Simplify.

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$p = - 3 \cdot \frac{2}{2} + \frac{9}{2}$

Combine $- 3$ and $\frac{2}{2}$ .

$p = \frac{- 3 \cdot 2}{2} + \frac{9}{2}$

Combine the numerators over the common denominator.

$p = \frac{- 3 \cdot 2 + 9}{2}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$p = \frac{- 6 + 9}{2}$

Add $- 6$ and $9$ .

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

Step 4

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

${(x - 4)}^{2} = 4 (\frac{3}{2}) (y + \frac{9}{2})$

Step 5

Simplify.

${(x - 4)}^{2} = 6 (y + \frac{9}{2})$

Step 6

Find the Parabola with Focus (4,-3) and Directrix y=-6 (4,-3) y=-6

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