Find the Directrix (y-3)^2=4(x-1)

${(y - 3)}^{2} = 4 (x - 1)$

Step 1

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

Rewrite the equation as $4 (x - 1) = {(y - 3)}^{2}$ .

$4 (x - 1) = {(y - 3)}^{2}$

Divide each term in $4 (x - 1) = {(y - 3)}^{2}$ by $4$ and simplify.

Divide each term in $4 (x - 1) = {(y - 3)}^{2}$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x - 1$ by $1$ .

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

Add $1$ to both sides of the equation.

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

Reorder terms.

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

Step 2

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

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

$h = 1$

$k = 3$

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.

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

Simplify.

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

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

Simplify by dividing numbers.

Divide $4$ by $4$ .

$\frac{1}{1}$

Divide $1$ 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.

$x = 0$

Step 6

Find the Directrix (y-3)^2=4(x-1)

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