Find the Focus y^2+8x-2y+41=0

$y^{2} + 8 x - 2 y + 41 = 0$

Step 1

Rewrite the equation in vertex form.

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

Move all terms not containing $x$ to the right side of the equation.

Subtract $y^{2}$ from both sides of the equation.

$8 x - 2 y + 41 = - y^{2}$

Add $2 y$ to both sides of the equation.

$8 x + 41 = - y^{2} + 2 y$

Subtract $41$ from both sides of the equation.

$8 x = - y^{2} + 2 y - 41$

Divide each term in $8 x = - y^{2} + 2 y - 41$ by $8$ and simplify.

Divide each term in $8 x = - y^{2} + 2 y - 41$ by $8$ .

$\frac{8 x}{8} = \frac{- y^{2}}{8} + \frac{2 y}{8} + \frac{- 41}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{- y^{2}}{8} + \frac{2 y}{8} + \frac{- 41}{8}$

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{8} + \frac{2 y}{8} + \frac{- 41}{8}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{8} + \frac{2 y}{8} + \frac{- 41}{8}$

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2 y$ .

$x = - \frac{y^{2}}{8} + \frac{2 (y)}{8} + \frac{- 41}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$x = - \frac{y^{2}}{8} + \frac{2 y}{2 \cdot 4} + \frac{- 41}{8}$

Cancel the common factor.

$x = - \frac{y^{2}}{8} + \frac{2 y}{2 \cdot 4} + \frac{- 41}{8}$

Rewrite the expression.

$x = - \frac{y^{2}}{8} + \frac{y}{4} + \frac{- 41}{8}$

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{8} + \frac{y}{4} - \frac{41}{8}$

Complete the square for $- \frac{y^{2}}{8} + \frac{y}{4} - \frac{41}{8}$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{1}{8}$

$b = \frac{1}{4}$

$c = - \frac{41}{8}$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{\frac{1}{4}}{2 (- \frac{1}{8})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{4} \cdot \frac{1}{2 (- \frac{1}{8})}$

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

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

$d = \frac{1}{4} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{8})}$

Move the negative in front of the fraction.

$d = \frac{1}{4} (- \frac{1}{2 (\frac{1}{8})})$

Combine $2$ and $\frac{1}{8}$ .

$d = \frac{1}{4} (- \frac{1}{\frac{2}{8}})$

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2$ .

$d = \frac{1}{4} (- \frac{1}{\frac{2 (1)}{8}})$

Cancel the common factors.

Factor $2$ out of $8$ .

$d = \frac{1}{4} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}})$

Cancel the common factor.

$d = \frac{1}{4} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}})$

Rewrite the expression.

$d = \frac{1}{4} (- \frac{1}{\frac{1}{4}})$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{4} (- (1 \cdot 4))$

Cancel the common factor of $4$ .

Factor $4$ out of $- (1 \cdot 4)$ .

$d = \frac{1}{4} (4 (- 1 \cdot (1)))$

Cancel the common factor.

$d = \frac{1}{4} (4 (- 1 \cdot (1)))$

Rewrite the expression.

$d = - 1 \cdot (1)$

Multiply $- 1$ by $1$ .

$d = - 1$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - \frac{41}{8} - \frac{{(\frac{1}{4})}^{2}}{4 (- \frac{1}{8})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{1}{4}$ .

$e = - \frac{41}{8} - \frac{\frac{1^{2}}{4^{2}}}{4 (- \frac{1}{8})}$

One to any power is one.

$e = - \frac{41}{8} - \frac{\frac{1}{4^{2}}}{4 (- \frac{1}{8})}$

Raise $4$ to the power of $2$ .

$e = - \frac{41}{8} - \frac{\frac{1}{16}}{4 (- \frac{1}{8})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - \frac{41}{8} - \frac{\frac{1}{16}}{- 4 (\frac{1}{8})}$

Combine $- 4$ and $\frac{1}{8}$ .

$e = - \frac{41}{8} - \frac{\frac{1}{16}}{\frac{- 4}{8}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $8$ .

Factor $4$ out of $- 4$ .

$e = - \frac{41}{8} - \frac{\frac{1}{16}}{\frac{4 (- 1)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$e = - \frac{41}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot - 1}{4 \cdot 2}}$

Cancel the common factor.

$e = - \frac{41}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot - 1}{4 \cdot 2}}$

Rewrite the expression.

$e = - \frac{41}{8} - \frac{\frac{1}{16}}{\frac{- 1}{2}}$

Move the negative in front of the fraction.

$e = - \frac{41}{8} - \frac{\frac{1}{16}}{- \frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{41}{8} - (\frac{1}{16} (- 1 \cdot 2))$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$e = - \frac{41}{8} - (\frac{1}{2 (8)} (- 1 \cdot 2))$

Factor $2$ out of $- 1 \cdot 2$ .

$e = - \frac{41}{8} - (\frac{1}{2 (8)} (2 \cdot - 1))$

Cancel the common factor.

$e = - \frac{41}{8} - (\frac{1}{2 \cdot 8} (2 \cdot - 1))$

Rewrite the expression.

$e = - \frac{41}{8} - (\frac{1}{8} \cdot - 1)$

Combine $\frac{1}{8}$ and $- 1$ .

$e = - \frac{41}{8} - \frac{- 1}{8}$

Move the negative in front of the fraction.

$e = - \frac{41}{8} - - \frac{1}{8}$

Multiply $- - \frac{1}{8}$ .

Multiply $- 1$ by $- 1$ .

$e = - \frac{41}{8} + 1 (\frac{1}{8})$

Multiply $\frac{1}{8}$ by $1$ .

$e = - \frac{41}{8} + \frac{1}{8}$

Combine the numerators over the common denominator.

$e = \frac{- 41 + 1}{8}$

Add $- 41$ and $1$ .

$e = \frac{- 40}{8}$

Divide $- 40$ by $8$ .

$e = - 5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{8} {(y - 1)}^{2} - 5$ .

$- \frac{1}{8} {(y - 1)}^{2} - 5$

Set $x$ equal to the new right side.

$x = - \frac{1}{8} \cdot {(y - 1)}^{2} - 5$

Step 2

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

$a = - \frac{1}{8}$

$h = - 5$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(- 5, 1)$

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.

$\frac{1}{4 \cdot (- \frac{1}{8})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{8})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{8})}$

Combine $4$ and $\frac{1}{8}$ .

$- \frac{1}{\frac{4}{8}}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 2)$

Multiply $- (1 \cdot 2)$ .

Multiply $2$ by $1$ .

$- 1 \cdot 2$

Multiply $- 1$ by $2$ .

$- 2$

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.

$(- 7, 1)$

Step 6

Find the Focus y^2+8x-2y+41=0

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