Find the Directrix (x+5)^2=12y

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

Step 1

Rewrite the equation in vertex form.

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

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

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

Divide each term in $12 y = {(x + 5)}^{2}$ by $12$ and simplify.

Divide each term in $12 y = {(x + 5)}^{2}$ by $12$ .

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

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Reorder terms.

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

Complete the square for $\frac{1}{12} \cdot {(x + 5)}^{2}$ .

Simplify the expression.

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$\frac{1}{12} \cdot ((x + 5) (x + 5))$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{12} \cdot (x (x + 5) + 5 (x + 5))$

Apply the distributive property.

$\frac{1}{12} \cdot (x \cdot x + x \cdot 5 + 5 (x + 5))$

Apply the distributive property.

$\frac{1}{12} \cdot (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{1}{12} \cdot (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5)$

Move $5$ to the left of $x$ .

$\frac{1}{12} \cdot (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5)$

Multiply $5$ by $5$ .

$\frac{1}{12} \cdot (x^{2} + 5 x + 5 x + 25)$

Add $5 x$ and $5 x$ .

$\frac{1}{12} \cdot (x^{2} + 10 x + 25)$

Apply the distributive property.

$\frac{1}{12} x^{2} + \frac{1}{12} (10 x) + \frac{1}{12} \cdot 25$

Simplify.

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

$\frac{x^{2}}{12} + \frac{1}{12} (10 x) + \frac{1}{12} \cdot 25$

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$\frac{x^{2}}{12} + \frac{1}{2 (6)} (10 x) + \frac{1}{12} \cdot 25$

Factor $2$ out of $10 x$ .

$\frac{x^{2}}{12} + \frac{1}{2 (6)} (2 (5 x)) + \frac{1}{12} \cdot 25$

Cancel the common factor.

$\frac{x^{2}}{12} + \frac{1}{2 \cdot 6} (2 (5 x)) + \frac{1}{12} \cdot 25$

Rewrite the expression.

$\frac{x^{2}}{12} + \frac{1}{6} (5 x) + \frac{1}{12} \cdot 25$

Combine $5$ and $\frac{1}{6}$ .

$\frac{x^{2}}{12} + \frac{5}{6} x + \frac{1}{12} \cdot 25$

Combine $\frac{5}{6}$ and $x$ .

$\frac{x^{2}}{12} + \frac{5 x}{6} + \frac{1}{12} \cdot 25$

Combine $\frac{1}{12}$ and $25$ .

$\frac{x^{2}}{12} + \frac{5 x}{6} + \frac{25}{12}$

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

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

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

$c = \frac{25}{12}$

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 $6$ .

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

$e = \frac{25}{12} - \frac{{(\frac{5}{6})}^{2}}{4 (\frac{1}{12})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

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

$e = \frac{25}{12} - \frac{\frac{5^{2}}{6^{2}}}{4 (\frac{1}{12})}$

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

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

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

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

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

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

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

Factor $4$ out of $4$ .

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

Cancel the common factors.

Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Factor $3$ out of $36$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Subtract $25$ from $25$ .

$e = \frac{0}{12}$

Divide $0$ by $12$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{12} {(x + 5)}^{2} + 0$ .

$\frac{1}{12} {(x + 5)}^{2} + 0$

Set $y$ equal to the new right side.

$y = \frac{1}{12} \cdot {(x + 5)}^{2} + 0$

Step 2

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

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

$h = - 5$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(- 5, 0)$

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 directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

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

$y = - 3$

Step 6

Find the Directrix (x+5)^2=12y

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