Find the Directrix y^2-6y+4x+9=0

$y^{2} - 6 y + 4 x + 9 = 0$

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.

$- 6 y + 4 x + 9 = - y^{2}$

Add $6 y$ to both sides of the equation.

$4 x + 9 = - y^{2} + 6 y$

Subtract $9$ from both sides of the equation.

$4 x = - y^{2} + 6 y - 9$

Divide each term in $4 x = - y^{2} + 6 y - 9$ by $4$ and simplify.

Divide each term in $4 x = - y^{2} + 6 y - 9$ by $4$ .

$\frac{4 x}{4} = \frac{- y^{2}}{4} + \frac{6 y}{4} + \frac{- 9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- y^{2}}{4} + \frac{6 y}{4} + \frac{- 9}{4}$

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{4} + \frac{6 y}{4} + \frac{- 9}{4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{4} + \frac{6 y}{4} + \frac{- 9}{4}$

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

Factor $2$ out of $6 y$ .

$x = - \frac{y^{2}}{4} + \frac{2 (3 y)}{4} + \frac{- 9}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = - \frac{y^{2}}{4} + \frac{2 (3 y)}{2 (2)} + \frac{- 9}{4}$

Cancel the common factor.

$x = - \frac{y^{2}}{4} + \frac{2 (3 y)}{2 \cdot 2} + \frac{- 9}{4}$

Rewrite the expression.

$x = - \frac{y^{2}}{4} + \frac{3 y}{2} + \frac{- 9}{4}$

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{4} + \frac{3 y}{2} - \frac{9}{4}$

Complete the square for $- \frac{y^{2}}{4} + \frac{3 y}{2} - \frac{9}{4}$ .

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

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

$b = \frac{3}{2}$

$c = - \frac{9}{4}$

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

$d = \frac{\frac{3}{2}}{2 (- \frac{1}{4})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{2} \cdot \frac{1}{2 (- \frac{1}{4})}$

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

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

$d = \frac{3}{2} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{4})}$

Move the negative in front of the fraction.

$d = \frac{3}{2} (- \frac{1}{2 (\frac{1}{4})})$

Combine $2$ and $\frac{1}{4}$ .

$d = \frac{3}{2} (- \frac{1}{\frac{2}{4}})$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$d = \frac{3}{2} (- \frac{1}{\frac{2 (1)}{4}})$

Cancel the common factors.

Factor $2$ out of $4$ .

$d = \frac{3}{2} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 2}})$

Cancel the common factor.

$d = \frac{3}{2} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 2}})$

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{2} (- (1 \cdot 2))$

Cancel the common factor of $2$ .

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

$d = \frac{3}{2} (2 (- 1 \cdot (1)))$

Cancel the common factor.

$d = \frac{3}{2} (2 (- 1 \cdot (1)))$

Rewrite the expression.

$d = 3 (- 1 \cdot (1))$

Multiply $- 1$ by $1$ .

$d = 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$d = - 3$

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

$e = - \frac{9}{4} - \frac{{(\frac{3}{2})}^{2}}{4 (- \frac{1}{4})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{3}{2}$ .

$e = - \frac{9}{4} - \frac{\frac{3^{2}}{2^{2}}}{4 (- \frac{1}{4})}$

Raise $3$ to the power of $2$ .

$e = - \frac{9}{4} - \frac{\frac{9}{2^{2}}}{4 (- \frac{1}{4})}$

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

$e = - \frac{9}{4} - \frac{\frac{9}{4}}{4 (- \frac{1}{4})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - \frac{9}{4} - \frac{\frac{9}{4}}{- 4 (\frac{1}{4})}$

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

$e = - \frac{9}{4} - \frac{\frac{9}{4}}{\frac{- 4}{4}}$

Divide $- 4$ by $4$ .

$e = - \frac{9}{4} - \frac{\frac{9}{4}}{- 1}$

Move the negative one from the denominator of $\frac{\frac{9}{4}}{- 1}$ .

$e = - \frac{9}{4} - (- 1 \cdot \frac{9}{4})$

Rewrite $- 1 \cdot \frac{9}{4}$ as $- \frac{9}{4}$ .

$e = - \frac{9}{4} - - \frac{9}{4}$

Multiply $- - \frac{9}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = - \frac{9}{4} + 1 (\frac{9}{4})$

Multiply $\frac{9}{4}$ by $1$ .

$e = - \frac{9}{4} + \frac{9}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 9 + 9}{4}$

Add $- 9$ and $9$ .

$e = \frac{0}{4}$

Divide $0$ by $4$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(y - 3)}^{2} + 0$ .

$- \frac{1}{4} {(y - 3)}^{2} + 0$

Set $x$ equal to the new right side.

$x = - \frac{1}{4} \cdot {(y - 3)}^{2} + 0$

Step 2

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

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

$h = 0$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(0, 3)$

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.

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

Simplify.

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

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

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

Move the negative in front of the fraction.

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

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

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

Divide $4$ by $4$ .

$- \frac{1}{1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$- \frac{1}{1}$

Rewrite the expression.

$- 1 \cdot 1$

Multiply $- 1$ by $1$ .

$- 1$

Step 5

Find the directrix.

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

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

$x = 1$

Step 6

Find the Directrix y^2-6y+4x+9=0

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