Pre-Algebra Examples

$x^{2} + 3 y^{2} - 12 y + 9 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Subtract $9$ from both sides of the equation.

$x^{2} + 3 y^{2} - 12 y = - 9$

Step 1.2

Complete the square for $3 y^{2} - 12 y$ .

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Step 1.2.1

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

$a = 3$

$b = - 12$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 12}{2 \cdot 3}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 12$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot 3}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 3$ .

$d = \frac{2 \cdot - 6}{2 (3)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot 3}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 6}{3}$

Step 1.2.3.2.2

Cancel the common factor of $- 6$ and $3$ .

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Step 1.2.3.2.2.1

Factor $3$ out of $- 6$ .

$d = \frac{3 \cdot - 2}{3}$

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $3$ out of $3$ .

$d = \frac{3 \cdot - 2}{3 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{3 \cdot - 2}{3 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 2}{1}$

Step 1.2.3.2.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 12)}^{2}}{4 \cdot 3}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raise $- 12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot 3}$

Step 1.2.4.2.1.2

Multiply $4$ by $3$ .

$e = 0 - \frac{144}{12}$

Step 1.2.4.2.1.3

Divide $144$ by $12$ .

$e = 0 - 1 \cdot 12$

Step 1.2.4.2.1.4

Multiply $- 1$ by $12$ .

$e = 0 - 12$

Step 1.2.4.2.2

Subtract $12$ from $0$ .

$e = - 12$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(y - 2)}^{2} - 12$ .

$3 {(y - 2)}^{2} - 12$

Step 1.3

Substitute $3 {(y - 2)}^{2} - 12$ for $3 y^{2} - 12 y$ in the equation $x^{2} + 3 y^{2} - 12 y = - 9$ .

$x^{2} + 3 {(y - 2)}^{2} - 12 = - 9$

Step 1.4

Move $- 12$ to the right side of the equation by adding $12$ to both sides.

$x^{2} + 3 {(y - 2)}^{2} = - 9 + 12$

Step 1.5

Add $- 9$ and $12$ .

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

Step 1.6

Divide each term by $3$ to make the right side equal to one.

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

Step 1.7

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

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

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

$\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = \sqrt{3}$

$b = 1$

$k = 2$

$h = 0$

Step 4

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(0, 2)$

Step 5

Find $c$ , the distance from the center to a focus.

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Step 5.1

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Step 5.2

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(\sqrt{3})}^{2} - {(1)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{{(3^{\frac{1}{2}})}^{2} - {(1)}^{2}}$

Step 5.3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{3^{\frac{1}{2} \cdot 2} - {(1)}^{2}}$

Step 5.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{3^{\frac{2}{2}} - {(1)}^{2}}$

Step 5.3.1.4

Cancel the common factor of $2$ .

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Step 5.3.1.4.1

Cancel the common factor.

$\sqrt{3^{\frac{2}{2}} - {(1)}^{2}}$

Step 5.3.1.4.2

Rewrite the expression.

$\sqrt{3^{1} - {(1)}^{2}}$

Step 5.3.1.5

Evaluate the exponent.

$\sqrt{3 - {(1)}^{2}}$

Step 5.3.2

Simplify the expression.

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Step 5.3.2.1

One to any power is one.

$\sqrt{3 - 1 \cdot 1}$

Step 5.3.2.2

Multiply $- 1$ by $1$ .

$\sqrt{3 - 1}$

Step 5.3.2.3

Subtract $1$ from $3$ .

$\sqrt{2}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding $a$ to $h$ .

$(h + a, k)$

Step 6.2

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

$(0 + \sqrt{3}, 2)$

Step 6.3

Simplify.

$(\sqrt{3}, 2)$

Step 6.4

The second vertex of an ellipse can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Step 6.5

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

$(0 - (\sqrt{3}), 2)$

Step 6.6

Simplify.

$(- \sqrt{3}, 2)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(\sqrt{3}, 2)$

${Vertex}_{2}$ : $(- \sqrt{3}, 2)$

${Vertex}_{1}$ : $(\sqrt{3}, 2)$

${Vertex}_{2}$ : $(- \sqrt{3}, 2)$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse can be found by adding $c$ to $h$ .

$(h + c, k)$

Step 7.2

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

$(0 + \sqrt{2}, 2)$

Step 7.3

Simplify.

$(\sqrt{2}, 2)$

Step 7.4

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Step 7.5

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

$(0 - (\sqrt{2}), 2)$

Step 7.6

Simplify.

$(- \sqrt{2}, 2)$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ : $(\sqrt{2}, 2)$

${Focus}_{2}$ : $(- \sqrt{2}, 2)$

${Focus}_{1}$ : $(\sqrt{2}, 2)$

${Focus}_{2}$ : $(- \sqrt{2}, 2)$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} - b^{2}}}{a}$

Step 8.2

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(\sqrt{3})}^{2} - {(1)}^{2}}}{\sqrt{3}}$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.3.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{{(3^{\frac{1}{2}})}^{2} - 1^{2}}}{\sqrt{3}}$

Step 8.3.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{3^{\frac{1}{2} \cdot 2} - 1^{2}}}{\sqrt{3}}$

Step 8.3.1.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{3^{\frac{2}{2}} - 1^{2}}}{\sqrt{3}}$

Step 8.3.1.1.4

Cancel the common factor of $2$ .

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Step 8.3.1.1.4.1

Cancel the common factor.

$\frac{\sqrt{3^{\frac{2}{2}} - 1^{2}}}{\sqrt{3}}$

Step 8.3.1.1.4.2

Rewrite the expression.

$\frac{\sqrt{3^{1} - 1^{2}}}{\sqrt{3}}$

Step 8.3.1.1.5

Evaluate the exponent.

$\frac{\sqrt{3 - 1^{2}}}{\sqrt{3}}$

Step 8.3.1.2

One to any power is one.

$\frac{\sqrt{3 - 1 \cdot 1}}{\sqrt{3}}$

Step 8.3.1.3

Multiply $- 1$ by $1$ .

$\frac{\sqrt{3 - 1}}{\sqrt{3}}$

Step 8.3.1.4

Subtract $1$ from $3$ .

$\frac{\sqrt{2}}{\sqrt{3}}$

Step 8.3.2

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 8.3.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 8.3.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 8.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 8.3.3.5

Add $1$ and $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 8.3.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 8.3.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 8.3.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.3.6.4

Cancel the common factor of $2$ .

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Step 8.3.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{2} \sqrt{3}}{3^{1}}$

Step 8.3.3.6.5

Evaluate the exponent.

$\frac{\sqrt{2} \sqrt{3}}{3}$

Step 8.3.4

Simplify the numerator.

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Step 8.3.4.1

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 3}}{3}$

Step 8.3.4.2

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{3}$

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(0, 2)$

${Vertex}_{1}$ : $(\sqrt{3}, 2)$

${Vertex}_{2}$ : $(- \sqrt{3}, 2)$

${Focus}_{1}$ : $(\sqrt{2}, 2)$

${Focus}_{2}$ : $(- \sqrt{2}, 2)$

Eccentricity: $\frac{\sqrt{6}}{3}$

Step 10