Find the Focus -12(y-6)^2=(x+5)

$- 12 {(y - 6)}^{2} = (x + 5)$

Step 1

Isolate $x$ to the left side of the equation.

Rewrite the equation as $x + 5 = - 12 {(y - 6)}^{2}$ .

$x + 5 = - 12 {(y - 6)}^{2}$

Subtract $5$ from both sides of the equation.

$x = - 12 {(y - 6)}^{2} - 5$

Step 2

Use the vertex form, $x = a {(y - k)}^{2} + h$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 12$

$h = - 5$

$k = 6$

Step 3

Find the vertex $(h, k)$ .

$(- 5, 6)$

Step 4

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 12}$

Simplify.

Multiply $4$ by $- 12$ .

$\frac{1}{- 48}$

Move the negative in front of the fraction.

$- \frac{1}{48}$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- \frac{241}{48}, 6)$

Step 6