Find the Focus (x-1)^2=-6(y-3)

${(x - 1)}^{2} = - 6 (y - 3)$

Step 1

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

Rewrite the equation as $- 6 (y - 3) = {(x - 1)}^{2}$ .

$- 6 (y - 3) = {(x - 1)}^{2}$

Divide each term in $- 6 (y - 3) = {(x - 1)}^{2}$ by $- 6$ and simplify.

Divide each term in $- 6 (y - 3) = {(x - 1)}^{2}$ by $- 6$ .

$\frac{- 6 (y - 3)}{- 6} = \frac{{(x - 1)}^{2}}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 (y - 3)}{- 6} = \frac{{(x - 1)}^{2}}{- 6}$

Divide $y - 3$ by $1$ .

$y - 3 = \frac{{(x - 1)}^{2}}{- 6}$

Simplify the right side.

Move the negative in front of the fraction.

$y - 3 = - \frac{{(x - 1)}^{2}}{6}$

Add $3$ to both sides of the equation.

$y = - \frac{{(x - 1)}^{2}}{6} + 3$

Reorder terms.

$y = - \frac{1}{6} \cdot {(x - 1)}^{2} + 3$

Step 2

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

$a = - \frac{1}{6}$

$h = 1$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(1, 3)$

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.

$\frac{1}{4 \cdot (- \frac{1}{6})}$

Simplify.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{6})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{6})}$

Combine $4$ and $\frac{1}{6}$ .

$- \frac{1}{\frac{4}{6}}$

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4$ .

$- \frac{1}{\frac{2 (2)}{6}}$

Cancel the common factors.

Factor $2$ out of $6$ .

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 3}}$

Cancel the common factor.

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 3}}$

Rewrite the expression.

$- \frac{1}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{3}{2}))$

Multiply $\frac{3}{2}$ by $1$ .

$- \frac{3}{2}$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

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

$(1, \frac{3}{2})$

Step 6

Find the Focus (x-1)^2=-6(y-3)

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