Find the Focus y^2=-1/5x

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

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $- \frac{1}{5} x = y^{2}$ .

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

Multiply both sides of the equation by $- 5$ .

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

Simplify the left side.

Simplify $- 5 (- \frac{1}{5} x)$ .

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

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

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{x}{5}$ into the numerator.

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

Factor $5$ out of $- 5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ 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 focus.

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

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

$(- \frac{1}{20}, 0)$

Step 6

Find the Focus y^2=-1/5x

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