Find the Directrix x^2=32y

$x^{2} = 32 y$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $32 y = x^{2}$ .

$32 y = x^{2}$

Divide each term in $32 y = x^{2}$ by $32$ and simplify.

Divide each term in $32 y = x^{2}$ by $32$ .

$\frac{32 y}{32} = \frac{x^{2}}{32}$

Simplify the left side.

Cancel the common factor of $32$ .

Cancel the common factor.

$\frac{32 y}{32} = \frac{x^{2}}{32}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{32}$

Complete the square for $\frac{x^{2}}{32}$ .

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

$a = \frac{1}{32}$

$b = 0$

$c = 0$

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

$d = \frac{0}{2 (\frac{1}{32})}$

Simplify the right side.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{1}{32})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{1}{32})}$

Rewrite the expression.

$d = \frac{0}{\frac{1}{32}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 \cdot 32$

Multiply $0$ by $32$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 (\frac{1}{32})}$

Simplify the right side.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$e = 0 - \frac{0}{4 (\frac{1}{32})}$

Combine $4$ and $\frac{1}{32}$ .

$e = 0 - \frac{0}{\frac{4}{32}}$

Cancel the common factor of $4$ and $32$ .

Factor $4$ out of $4$ .

$e = 0 - \frac{0}{\frac{4 (1)}{32}}$

Cancel the common factors.

Factor $4$ out of $32$ .

$e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 8}}$

Cancel the common factor.

$e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 8}}$

Rewrite the expression.

$e = 0 - \frac{0}{\frac{1}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 \cdot 8)$

Multiply $- (0 \cdot 8)$ .

Multiply $0$ by $8$ .

$e = 0 - 0$

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Add $0$ and $0$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{32} x^{2}$ .

$\frac{1}{32} x^{2}$

Set $y$ equal to the new right side.

$y = \frac{1}{32} x^{2}$

Step 2

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

$a = \frac{1}{32}$

$h = 0$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(0, 0)$

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.

$\frac{1}{4 \cdot \frac{1}{32}}$

Simplify.

Combine $4$ and $\frac{1}{32}$ .

$\frac{1}{\frac{4}{32}}$

Cancel the common factor of $4$ and $32$ .

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{32}}$

Cancel the common factors.

Factor $4$ out of $32$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 8}}$

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 8}}$

Rewrite the expression.

$\frac{1}{\frac{1}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 8$

Multiply $8$ by $1$ .

$8$

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.

$y = - 8$

Step 6

Find the Directrix x^2=32y

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