Find the Focus 9x^2-2y=0

$9 x^{2} - 2 y = 0$

Step 1

Rewrite the equation in vertex form.

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

Subtract $9 x^{2}$ from both sides of the equation.

$- 2 y = - 9 x^{2}$

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

$y = \frac{9 x^{2}}{2}$

Complete the square for $\frac{9 x^{2}}{2}$ .

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

$a = \frac{9}{2}$

$b = 0$

$c = 0$

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

$d = \frac{0}{2 (\frac{9}{2})}$

Simplify the right side.

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

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{9}{2})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{9}{2})}$

Rewrite the expression.

$d = \frac{0}{\frac{9}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (\frac{2}{9})$

Multiply $0$ by $\frac{2}{9}$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 (\frac{9}{2})}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{0}{4 (\frac{9}{2})}$

Combine $4$ and $\frac{9}{2}$ .

$e = 0 - \frac{0}{\frac{4 \cdot 9}{2}}$

Multiply $4$ by $9$ .

$e = 0 - \frac{0}{\frac{36}{2}}$

Divide $36$ by $2$ .

$e = 0 - \frac{0}{18}$

Divide $0$ by $18$ .

$e = 0 - 0$

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 $\frac{9}{2} x^{2}$ .

$\frac{9}{2} x^{2}$

Set $y$ equal to the new right side.

$y = \frac{9}{2} x^{2}$

Step 2

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

$a = \frac{9}{2}$

$h = 0$

$k = 0$

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.

$\frac{1}{4 \cdot \frac{9}{2}}$

Simplify.

Combine $4$ and $\frac{9}{2}$ .

$\frac{1}{\frac{4 \cdot 9}{2}}$

Simplify the expression.

Multiply $4$ by $9$ .

$\frac{1}{\frac{36}{2}}$

Divide $36$ by $2$ .

$\frac{1}{18}$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

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

$(0, \frac{1}{18})$

Step 6

Find the Focus 9x^2-2y=0

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