Find the Focus x^2-2x+16y-47=0

$x^{2} - 2 x + 16 y - 47 = 0$

Step 1

Rewrite the equation in vertex form.

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

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

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

$- 2 x + 16 y - 47 = - x^{2}$

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

$16 y - 47 = - x^{2} + 2 x$

Add $47$ to both sides of the equation.

$16 y = - x^{2} + 2 x + 47$

Divide each term in $16 y = - x^{2} + 2 x + 47$ by $16$ and simplify.

Divide each term in $16 y = - x^{2} + 2 x + 47$ by $16$ .

$\frac{16 y}{16} = \frac{- x^{2}}{16} + \frac{2 x}{16} + \frac{47}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 y}{16} = \frac{- x^{2}}{16} + \frac{2 x}{16} + \frac{47}{16}$

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{16} + \frac{2 x}{16} + \frac{47}{16}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{16} + \frac{2 x}{16} + \frac{47}{16}$

Cancel the common factor of $2$ and $16$ .

Factor $2$ out of $2 x$ .

$y = - \frac{x^{2}}{16} + \frac{2 (x)}{16} + \frac{47}{16}$

Cancel the common factors.

Factor $2$ out of $16$ .

$y = - \frac{x^{2}}{16} + \frac{2 x}{2 \cdot 8} + \frac{47}{16}$

Cancel the common factor.

$y = - \frac{x^{2}}{16} + \frac{2 x}{2 \cdot 8} + \frac{47}{16}$

Rewrite the expression.

$y = - \frac{x^{2}}{16} + \frac{x}{8} + \frac{47}{16}$

Complete the square for $- \frac{x^{2}}{16} + \frac{x}{8} + \frac{47}{16}$ .

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

$a = - \frac{1}{16}$

$b = \frac{1}{8}$

$c = \frac{47}{16}$

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 $d = \frac{b}{2 a}$ .

$d = \frac{\frac{1}{8}}{2 (- \frac{1}{16})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Move the negative in front of the fraction.

$d = \frac{1}{8} (- \frac{1}{2 (\frac{1}{16})})$

Combine $2$ and $\frac{1}{16}$ .

$d = \frac{1}{8} (- \frac{1}{\frac{2}{16}})$

Cancel the common factor of $2$ and $16$ .

Factor $2$ out of $2$ .

$d = \frac{1}{8} (- \frac{1}{\frac{2 (1)}{16}})$

Cancel the common factors.

Factor $2$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $8$ .

Factor $8$ out of $- (1 \cdot 8)$ .

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

Cancel the common factor.

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

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{47}{16} - \frac{{(\frac{1}{8})}^{2}}{4 (- \frac{1}{16})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{1}{8}$ .

$e = \frac{47}{16} - \frac{\frac{1^{2}}{8^{2}}}{4 (- \frac{1}{16})}$

One to any power is one.

$e = \frac{47}{16} - \frac{\frac{1}{8^{2}}}{4 (- \frac{1}{16})}$

Raise $8$ to the power of $2$ .

$e = \frac{47}{16} - \frac{\frac{1}{64}}{4 (- \frac{1}{16})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = \frac{47}{16} - \frac{\frac{1}{64}}{- 4 (\frac{1}{16})}$

Combine $- 4$ and $\frac{1}{16}$ .

$e = \frac{47}{16} - \frac{\frac{1}{64}}{\frac{- 4}{16}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $16$ .

Factor $4$ out of $- 4$ .

$e = \frac{47}{16} - \frac{\frac{1}{64}}{\frac{4 (- 1)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$e = \frac{47}{16} - \frac{\frac{1}{64}}{\frac{4 \cdot - 1}{4 \cdot 4}}$

Cancel the common factor.

$e = \frac{47}{16} - \frac{\frac{1}{64}}{\frac{4 \cdot - 1}{4 \cdot 4}}$

Rewrite the expression.

$e = \frac{47}{16} - \frac{\frac{1}{64}}{\frac{- 1}{4}}$

Move the negative in front of the fraction.

$e = \frac{47}{16} - \frac{\frac{1}{64}}{- \frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{47}{16} - (\frac{1}{64} (- 1 \cdot 4))$

Cancel the common factor of $4$ .

Factor $4$ out of $64$ .

$e = \frac{47}{16} - (\frac{1}{4 (16)} (- 1 \cdot 4))$

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

$e = \frac{47}{16} - (\frac{1}{4 (16)} (4 \cdot - 1))$

Cancel the common factor.

$e = \frac{47}{16} - (\frac{1}{4 \cdot 16} (4 \cdot - 1))$

Rewrite the expression.

$e = \frac{47}{16} - (\frac{1}{16} \cdot - 1)$

Combine $\frac{1}{16}$ and $- 1$ .

$e = \frac{47}{16} - \frac{- 1}{16}$

Move the negative in front of the fraction.

$e = \frac{47}{16} - - \frac{1}{16}$

Multiply $- - \frac{1}{16}$ .

Multiply $- 1$ by $- 1$ .

$e = \frac{47}{16} + 1 (\frac{1}{16})$

Multiply $\frac{1}{16}$ by $1$ .

$e = \frac{47}{16} + \frac{1}{16}$

Combine the numerators over the common denominator.

$e = \frac{47 + 1}{16}$

Add $47$ and $1$ .

$e = \frac{48}{16}$

Divide $48$ by $16$ .

$e = 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{16} {(x - 1)}^{2} + 3$ .

$- \frac{1}{16} {(x - 1)}^{2} + 3$

Set $y$ equal to the new right side.

$y = - \frac{1}{16} \cdot {(x - 1)}^{2} + 3$

Step 2

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

$a = - \frac{1}{16}$

$h = 1$

$k = 3$

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.

$\frac{1}{4 \cdot (- \frac{1}{16})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{16})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{16})}$

Combine $4$ and $\frac{1}{16}$ .

$- \frac{1}{\frac{4}{16}}$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 4)$

Multiply $- (1 \cdot 4)$ .

Multiply $4$ by $1$ .

$- 1 \cdot 4$

Multiply $- 1$ by $4$ .

$- 4$

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.

$(1, - 1)$

Step 6

Find the Focus x^2-2x+16y-47=0

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