Find the Directrix -5y^2-x=0

$- 5 y^{2} - x = 0$

Step 1

Rewrite the equation in vertex form.

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

Add $5 y^{2}$ to both sides of the equation.

$- x = 5 y^{2}$

Divide each term in $- x = 5 y^{2}$ by $- 1$ and simplify.

Divide each term in $- x = 5 y^{2}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{5 y^{2}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{5 y^{2}}{- 1}$

Divide $x$ by $1$ .

$x = \frac{5 y^{2}}{- 1}$

Simplify the right side.

Move the negative one from the denominator of $\frac{5 y^{2}}{- 1}$ .

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

Rewrite $- 1 \cdot (5 y^{2})$ as $- (5 y^{2})$ .

$x = - (5 y^{2})$

Multiply $5$ by $- 1$ .

$x = - 5 y^{2}$

Complete the square for $- 5 y^{2}$ .

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

$a = - 5$

$b = 0$

$c = 0$

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

$d = \frac{0}{2 \cdot - 5}$

Simplify the right side.

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

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot - 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

$d = \frac{2 (0)}{2 (- 5)}$

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot - 5}$

Rewrite the expression.

$d = \frac{0}{- 5}$

Cancel the common factor of $0$ and $- 5$ .

Factor $5$ out of $0$ .

$d = \frac{5 (0)}{- 5}$

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Multiply $- 1$ by $0$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 \cdot - 5}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{0}{4 \cdot - 5}$

Multiply $4$ by $- 5$ .

$e = 0 - \frac{0}{- 20}$

Divide $0$ by $- 20$ .

$e = 0 - 0$

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 $- 5 y^{2}$ .

$- 5 y^{2}$

Set $x$ equal to the new right side.

$x = - 5 y^{2}$

Step 2

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

$a = - 5$

$h = 0$

$k = 0$

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.

$\frac{1}{4 \cdot - 5}$

Simplify.

Multiply $4$ by $- 5$ .

$\frac{1}{- 20}$

Move the negative in front of the fraction.

$- \frac{1}{20}$

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.

$x = \frac{1}{20}$

Step 6

Find the Directrix -5y^2-x=0

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