Find the Parabola with Focus (6,-4) and Directrix y=-7 (6,-4) y=-7

$(6, - 4)$ $y = - 7$

Since the directrix is vertical, use the equation of a parabola that opens up or down.

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

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.

$(6, \frac{- 4 - 7}{2})$

Simplify the vertex.

Subtract $7$ from $- 4$ .

$(6, \frac{- 11}{2})$

Move the negative in front of the fraction.

$(6, - \frac{11}{2})$

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 $p$ .

$p = - 4 + \frac{11}{2}$

Simplify.

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$p = - 4 \cdot \frac{2}{2} + \frac{11}{2}$

Combine $- 4$ and $\frac{2}{2}$ .

$p = \frac{- 4 \cdot 2}{2} + \frac{11}{2}$

Combine the numerators over the common denominator.

$p = \frac{- 4 \cdot 2 + 11}{2}$

Simplify the numerator.

Multiply $- 4$ by $2$ .

$p = \frac{- 8 + 11}{2}$

Add $- 8$ and $11$ .

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

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

${(x - 6)}^{2} = 4 (\frac{3}{2}) (y + \frac{11}{2})$

Simplify.

${(x - 6)}^{2} = 6 (y + \frac{11}{2})$

Find the Parabola with Focus (6,-4) and Directrix y=-7 (6,-4) y=-7

Related Questions

Support LCMGCF.com by disabling your adblocker.