Find the Focus x^2-10x-12y-23=0

$x^{2} - 10 x - 12 y - 23 = 0$

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.

$- 10 x - 12 y - 23 = - x^{2}$

Add $10 x$ to both sides of the equation.

$- 12 y - 23 = - x^{2} + 10 x$

Add $23$ to both sides of the equation.

$- 12 y = - x^{2} + 10 x + 23$

Divide each term in $- 12 y = - x^{2} + 10 x + 23$ by $- 12$ and simplify.

Divide each term in $- 12 y = - x^{2} + 10 x + 23$ by $- 12$ .

$\frac{- 12 y}{- 12} = \frac{- x^{2}}{- 12} + \frac{10 x}{- 12} + \frac{23}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 y}{- 12} = \frac{- x^{2}}{- 12} + \frac{10 x}{- 12} + \frac{23}{- 12}$

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 12} + \frac{10 x}{- 12} + \frac{23}{- 12}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{12} + \frac{10 x}{- 12} + \frac{23}{- 12}$

Cancel the common factor of $10$ and $- 12$ .

Factor $2$ out of $10 x$ .

$y = \frac{x^{2}}{12} + \frac{2 (5 x)}{- 12} + \frac{23}{- 12}$

Cancel the common factors.

Factor $2$ out of $- 12$ .

$y = \frac{x^{2}}{12} + \frac{2 (5 x)}{2 (- 6)} + \frac{23}{- 12}$

Cancel the common factor.

$y = \frac{x^{2}}{12} + \frac{2 (5 x)}{2 \cdot - 6} + \frac{23}{- 12}$

Rewrite the expression.

$y = \frac{x^{2}}{12} + \frac{5 x}{- 6} + \frac{23}{- 12}$

Move the negative in front of the fraction.

$y = \frac{x^{2}}{12} - \frac{5 x}{6} + \frac{23}{- 12}$

Move the negative in front of the fraction.

$y = \frac{x^{2}}{12} - \frac{5 x}{6} - \frac{23}{12}$

Complete the square for $\frac{x^{2}}{12} - \frac{5 x}{6} - \frac{23}{12}$ .

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

$a = \frac{1}{12}$

$b = - \frac{5}{6}$

$c = - \frac{23}{12}$

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

$d = \frac{- \frac{5}{6}}{2 (\frac{1}{12})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{5}{6} \cdot \frac{1}{2 (\frac{1}{12})}$

Combine $2$ and $\frac{1}{12}$ .

$d = - \frac{5}{6} \cdot \frac{1}{\frac{2}{12}}$

Cancel the common factor of $2$ and $12$ .

Factor $2$ out of $2$ .

$d = - \frac{5}{6} \cdot \frac{1}{\frac{2 (1)}{12}}$

Cancel the common factors.

Factor $2$ out of $12$ .

$d = - \frac{5}{6} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Cancel the common factor.

$d = - \frac{5}{6} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Rewrite the expression.

$d = - \frac{5}{6} \cdot \frac{1}{\frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{5}{6} (1 \cdot 6)$

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$d = \frac{- 5}{6} (1 \cdot 6)$

Factor $6$ out of $1 \cdot 6$ .

$d = \frac{- 5}{6} (6 \cdot 1)$

Cancel the common factor.

$d = \frac{- 5}{6} (6 \cdot 1)$

Rewrite the expression.

$d = - 5$

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

$e = - \frac{23}{12} - \frac{{(- \frac{5}{6})}^{2}}{4 (\frac{1}{12})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{5}{6}$ .

$e = - \frac{23}{12} - \frac{{(- 1)}^{2} {(\frac{5}{6})}^{2}}{4 (\frac{1}{12})}$

Raise $- 1$ to the power of $2$ .

$e = - \frac{23}{12} - \frac{1 {(\frac{5}{6})}^{2}}{4 (\frac{1}{12})}$

Apply the product rule to $\frac{5}{6}$ .

$e = - \frac{23}{12} - \frac{1 \frac{5^{2}}{6^{2}}}{4 (\frac{1}{12})}$

Raise $5$ to the power of $2$ .

$e = - \frac{23}{12} - \frac{1 \frac{25}{6^{2}}}{4 (\frac{1}{12})}$

Raise $6$ to the power of $2$ .

$e = - \frac{23}{12} - \frac{1 (\frac{25}{36})}{4 (\frac{1}{12})}$

Multiply $\frac{25}{36}$ by $1$ .

$e = - \frac{23}{12} - \frac{\frac{25}{36}}{4 (\frac{1}{12})}$

Combine $4$ and $\frac{1}{12}$ .

$e = - \frac{23}{12} - \frac{\frac{25}{36}}{\frac{4}{12}}$

Cancel the common factor of $4$ and $12$ .

Factor $4$ out of $4$ .

$e = - \frac{23}{12} - \frac{\frac{25}{36}}{\frac{4 (1)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

$e = - \frac{23}{12} - \frac{\frac{25}{36}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Cancel the common factor.

$e = - \frac{23}{12} - \frac{\frac{25}{36}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Rewrite the expression.

$e = - \frac{23}{12} - \frac{\frac{25}{36}}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{23}{12} - (\frac{25}{36} \cdot 3)$

Cancel the common factor of $3$ .

Factor $3$ out of $36$ .

$e = - \frac{23}{12} - (\frac{25}{3 (12)} \cdot 3)$

Cancel the common factor.

$e = - \frac{23}{12} - (\frac{25}{3 \cdot 12} \cdot 3)$

Rewrite the expression.

$e = - \frac{23}{12} - \frac{25}{12}$

Combine the numerators over the common denominator.

$e = \frac{- 23 - 25}{12}$

Subtract $25$ from $- 23$ .

$e = \frac{- 48}{12}$

Divide $- 48$ by $12$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{12} {(x - 5)}^{2} - 4$ .

$\frac{1}{12} {(x - 5)}^{2} - 4$

Set $y$ equal to the new right side.

$y = \frac{1}{12} \cdot {(x - 5)}^{2} - 4$

Step 2

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

$a = \frac{1}{12}$

$h = 5$

$k = - 4$

Step 3

Find the vertex $(h, k)$ .

$(5, - 4)$

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.

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

Simplify.

Combine $4$ and $\frac{1}{12}$ .

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

Cancel the common factor of $4$ and $12$ .

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 3}}$

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 3}}$

Rewrite the expression.

$\frac{1}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 3$

Multiply $3$ by $1$ .

$3$

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.

$(5, - 1)$

Step 6

Find the Focus x^2-10x-12y-23=0

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