Precalculus Examples

$4 x^{2} + 3 y^{2} + 8 x - 24 y + 51 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Subtract $51$ from both sides of the equation.

$4 x^{2} + 3 y^{2} + 8 x - 24 y = - 51$

Step 1.2

Complete the square for $4 x^{2} + 8 x$ .

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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 $c$ .

$a = 4$

$b = 8$

$c = 0$

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

$d = \frac{8}{2 \cdot 4}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $8$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2 \cdot 4}$

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 $2 \cdot 4$ .

$d = \frac{2 \cdot 4}{2 (4)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot 4}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{4}{4}$

Step 1.2.3.2.2

Cancel the common factor of $4$ .

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Step 1.2.3.2.2.1

Cancel the common factor.

$d = \frac{4}{4}$

Step 1.2.3.2.2.2

Rewrite the expression.

$d = 1$

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

$e = 0 - \frac{8^{2}}{4 \cdot 4}$

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 $8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 4}$

Step 1.2.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{64}{16}$

Step 1.2.4.2.1.3

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Step 1.2.4.2.1.4

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 1.2.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + 1)}^{2} - 4$ .

$4 {(x + 1)}^{2} - 4$

Step 1.3

Substitute $4 {(x + 1)}^{2} - 4$ for $4 x^{2} + 8 x$ in the equation $4 x^{2} + 3 y^{2} + 8 x - 24 y = - 51$ .

$4 {(x + 1)}^{2} - 4 + 3 y^{2} - 24 y = - 51$

Step 1.4

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

$4 {(x + 1)}^{2} + 3 y^{2} - 24 y = - 51 + 4$

Step 1.5

Complete the square for $3 y^{2} - 24 y$ .

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Step 1.5.1

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

$a = 3$

$b = - 24$

$c = 0$

Step 1.5.2

Consider the vertex form of a parabola.

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

Step 1.5.3

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

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Step 1.5.3.1

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

$d = \frac{- 24}{2 \cdot 3}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $- 24$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $- 24$ .

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

Step 1.5.3.2.1.2

Cancel the common factors.

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Step 1.5.3.2.1.2.1

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

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

Step 1.5.3.2.1.2.2

Cancel the common factor.

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

Step 1.5.3.2.1.2.3

Rewrite the expression.

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

Step 1.5.3.2.2

Cancel the common factor of $- 12$ and $3$ .

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Step 1.5.3.2.2.1

Factor $3$ out of $- 12$ .

$d = \frac{3 \cdot - 4}{3}$

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $3$ out of $3$ .

$d = \frac{3 \cdot - 4}{3 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{3 \cdot - 4}{3 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 4}{1}$

Step 1.5.3.2.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

Step 1.5.4

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

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Step 1.5.4.1

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

$e = 0 - \frac{{(- 24)}^{2}}{4 \cdot 3}$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Simplify each term.

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Step 1.5.4.2.1.1

Raise $- 24$ to the power of $2$ .

$e = 0 - \frac{576}{4 \cdot 3}$

Step 1.5.4.2.1.2

Multiply $4$ by $3$ .

$e = 0 - \frac{576}{12}$

Step 1.5.4.2.1.3

Divide $576$ by $12$ .

$e = 0 - 1 \cdot 48$

Step 1.5.4.2.1.4

Multiply $- 1$ by $48$ .

$e = 0 - 48$

Step 1.5.4.2.2

Subtract $48$ from $0$ .

$e = - 48$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(y - 4)}^{2} - 48$ .

$3 {(y - 4)}^{2} - 48$

Step 1.6

Substitute $3 {(y - 4)}^{2} - 48$ for $3 y^{2} - 24 y$ in the equation $4 x^{2} + 3 y^{2} + 8 x - 24 y = - 51$ .

$4 {(x + 1)}^{2} + 3 {(y - 4)}^{2} - 48 = - 51 + 4$

Step 1.7

Move $- 48$ to the right side of the equation by adding $48$ to both sides.

$4 {(x + 1)}^{2} + 3 {(y - 4)}^{2} = - 51 + 4 + 48$

Step 1.8

Simplify $- 51 + 4 + 48$ .

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Step 1.8.1

Add $- 51$ and $4$ .

$4 {(x + 1)}^{2} + 3 {(y - 4)}^{2} = - 47 + 48$

Step 1.8.2

Add $- 47$ and $48$ .

$4 {(x + 1)}^{2} + 3 {(y - 4)}^{2} = 1$

Step 1.9

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 $1$ .

$\frac{{(x + 1)}^{2}}{\frac{1}{4}} + \frac{{(y - 4)}^{2}}{\frac{1}{3}} = 1$

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.

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

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.

$a = \frac{\sqrt{3}}{3}$

$b = \frac{1}{2}$

$k = 4$

$h = - 1$

Step 4

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

$(- 1, 4)$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

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

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

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

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

Step 5.3.2.3

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

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

Step 5.3.2.4

Cancel the common factor of $2$ .

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Step 5.3.2.4.1

Cancel the common factor.

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

Step 5.3.2.4.2

Rewrite the expression.

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

Step 5.3.2.5

Evaluate the exponent.

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

Step 5.3.3

Raise $3$ to the power of $2$ .

$\sqrt{\frac{3}{9} - {(\frac{1}{2})}^{2}}$

Step 5.3.4

Cancel the common factor of $3$ and $9$ .

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Step 5.3.4.1

Factor $3$ out of $3$ .

$\sqrt{\frac{3 (1)}{9} - {(\frac{1}{2})}^{2}}$

Step 5.3.4.2

Cancel the common factors.

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Step 5.3.4.2.1

Factor $3$ out of $9$ .

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

Step 5.3.4.2.2

Cancel the common factor.

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

Step 5.3.4.2.3

Rewrite the expression.

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

Step 5.3.5

Simplify the expression.

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Step 5.3.5.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 5.3.5.2

One to any power is one.

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

Step 5.3.5.3

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

$\sqrt{\frac{1}{3} - \frac{1}{4}}$

Step 5.3.6

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{1}{3} \cdot \frac{4}{4} - \frac{1}{4}}$

Step 5.3.7

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{\frac{1}{3} \cdot \frac{4}{4} - \frac{1}{4} \cdot \frac{3}{3}}$

Step 5.3.8

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.8.1

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$\sqrt{\frac{4}{3 \cdot 4} - \frac{1}{4} \cdot \frac{3}{3}}$

Step 5.3.8.2

Multiply $3$ by $4$ .

$\sqrt{\frac{4}{12} - \frac{1}{4} \cdot \frac{3}{3}}$

Step 5.3.8.3

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\sqrt{\frac{4}{12} - \frac{3}{4 \cdot 3}}$

Step 5.3.8.4

Multiply $4$ by $3$ .

$\sqrt{\frac{4}{12} - \frac{3}{12}}$

Step 5.3.9

Combine the numerators over the common denominator.

$\sqrt{\frac{4 - 3}{12}}$

Step 5.3.10

Subtract $3$ from $4$ .

$\sqrt{\frac{1}{12}}$

Step 5.3.11

Rewrite $\sqrt{\frac{1}{12}}$ as $\frac{\sqrt{1}}{\sqrt{12}}$ .

$\frac{\sqrt{1}}{\sqrt{12}}$

Step 5.3.12

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{12}}$

Step 5.3.13

Simplify the denominator.

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Step 5.3.13.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.3.13.1.1

Factor $4$ out of $12$ .

$\frac{1}{\sqrt{4 (3)}}$

Step 5.3.13.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 5.3.13.2

Pull terms out from under the radical.

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

Step 5.3.14

Multiply $\frac{1}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.3.15

Combine and simplify the denominator.

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Step 5.3.15.1

Multiply $\frac{1}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.3.15.2

Move $\sqrt{3}$ .

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

Step 5.3.15.3

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

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

Step 5.3.15.4

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

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

Step 5.3.15.5

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

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

Step 5.3.15.6

Add $1$ and $1$ .

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

Step 5.3.15.7

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

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Step 5.3.15.7.1

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

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

Step 5.3.15.7.2

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

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

Step 5.3.15.7.3

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

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

Step 5.3.15.7.4

Cancel the common factor of $2$ .

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Step 5.3.15.7.4.1

Cancel the common factor.

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

Step 5.3.15.7.4.2

Rewrite the expression.

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

Step 5.3.15.7.5

Evaluate the exponent.

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

Step 5.3.16

Multiply $2$ by $3$ .

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

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 $k$ .

$(h, k + a)$

Step 6.2

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

$(- 1, 4 + \frac{\sqrt{3}}{3})$

Step 6.3

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

$(h, k - a)$

Step 6.4

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

$(- 1, 4 - (\frac{\sqrt{3}}{3}))$

Step 6.5

Simplify.

$(- 1, 4 - \frac{\sqrt{3}}{3})$

Step 6.6

Ellipses have two vertices.

${Vertex}_{1}$ : $(- 1, 4 + \frac{\sqrt{3}}{3})$

${Vertex}_{2}$ : $(- 1, 4 - \frac{\sqrt{3}}{3})$

${Vertex}_{1}$ : $(- 1, 4 + \frac{\sqrt{3}}{3})$

${Vertex}_{2}$ : $(- 1, 4 - \frac{\sqrt{3}}{3})$

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 $k$ .

$(h, k + c)$

Step 7.2

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

$(- 1, 4 + \frac{\sqrt{3}}{6})$

Step 7.3

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 7.4

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

$(- 1, 4 - (\frac{\sqrt{3}}{6}))$

Step 7.5

Simplify.

$(- 1, 4 - \frac{\sqrt{3}}{6})$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(- 1, 4 + \frac{\sqrt{3}}{6})$

${Focus}_{2}$ : $(- 1, 4 - \frac{\sqrt{3}}{6})$

${Focus}_{1}$ : $(- 1, 4 + \frac{\sqrt{3}}{6})$

${Focus}_{2}$ : $(- 1, 4 - \frac{\sqrt{3}}{6})$

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.

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

Step 8.3

Simplify.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.3.2

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

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

Step 8.3.3

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

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Step 8.3.3.1

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

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

Step 8.3.3.2

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

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

Step 8.3.3.3

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

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

Step 8.3.3.4

Cancel the common factor of $2$ .

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Step 8.3.3.4.1

Cancel the common factor.

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

Step 8.3.3.4.2

Rewrite the expression.

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

Step 8.3.3.5

Evaluate the exponent.

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

Step 8.3.4

Raise $3$ to the power of $2$ .

$\sqrt{\frac{3}{9} - {(\frac{1}{2})}^{2}} \frac{3}{\sqrt{3}}$

Step 8.3.5

Cancel the common factor of $3$ and $9$ .

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Step 8.3.5.1

Factor $3$ out of $3$ .

$\sqrt{\frac{3 (1)}{9} - {(\frac{1}{2})}^{2}} \frac{3}{\sqrt{3}}$

Step 8.3.5.2

Cancel the common factors.

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Step 8.3.5.2.1

Factor $3$ out of $9$ .

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

Step 8.3.5.2.2

Cancel the common factor.

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

Step 8.3.5.2.3

Rewrite the expression.

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

Step 8.3.6

Simplify the expression.

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Step 8.3.6.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 8.3.6.2

One to any power is one.

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

Step 8.3.6.3

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

$\sqrt{\frac{1}{3} - \frac{1}{4}} \frac{3}{\sqrt{3}}$

Step 8.3.7

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{1}{3} \cdot \frac{4}{4} - \frac{1}{4}} \frac{3}{\sqrt{3}}$

Step 8.3.8

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{\frac{1}{3} \cdot \frac{4}{4} - \frac{1}{4} \cdot \frac{3}{3}} \frac{3}{\sqrt{3}}$

Step 8.3.9

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.9.1

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$\sqrt{\frac{4}{3 \cdot 4} - \frac{1}{4} \cdot \frac{3}{3}} \frac{3}{\sqrt{3}}$

Step 8.3.9.2

Multiply $3$ by $4$ .

$\sqrt{\frac{4}{12} - \frac{1}{4} \cdot \frac{3}{3}} \frac{3}{\sqrt{3}}$

Step 8.3.9.3

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\sqrt{\frac{4}{12} - \frac{3}{4 \cdot 3}} \frac{3}{\sqrt{3}}$

Step 8.3.9.4

Multiply $4$ by $3$ .

$\sqrt{\frac{4}{12} - \frac{3}{12}} \frac{3}{\sqrt{3}}$

Step 8.3.10

Combine the numerators over the common denominator.

$\sqrt{\frac{4 - 3}{12}} \frac{3}{\sqrt{3}}$

Step 8.3.11

Subtract $3$ from $4$ .

$\sqrt{\frac{1}{12}} \frac{3}{\sqrt{3}}$

Step 8.3.12

Rewrite $\sqrt{\frac{1}{12}}$ as $\frac{\sqrt{1}}{\sqrt{12}}$ .

$\frac{\sqrt{1}}{\sqrt{12}} \cdot \frac{3}{\sqrt{3}}$

Step 8.3.13

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{12}} \cdot \frac{3}{\sqrt{3}}$

Step 8.3.14

Simplify the denominator.

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Step 8.3.14.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.3.14.1.1

Factor $4$ out of $12$ .

$\frac{1}{\sqrt{4 (3)}} \cdot \frac{3}{\sqrt{3}}$

Step 8.3.14.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 8.3.14.2

Pull terms out from under the radical.

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

Step 8.3.15

Multiply $\frac{1}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 8.3.16

Combine and simplify the denominator.

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Step 8.3.16.1

Multiply $\frac{1}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 8.3.16.2

Move $\sqrt{3}$ .

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

Step 8.3.16.3

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

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

Step 8.3.16.4

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

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

Step 8.3.16.5

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

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

Step 8.3.16.6

Add $1$ and $1$ .

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

Step 8.3.16.7

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

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Step 8.3.16.7.1

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

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

Step 8.3.16.7.2

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

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

Step 8.3.16.7.3

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

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

Step 8.3.16.7.4

Cancel the common factor of $2$ .

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Step 8.3.16.7.4.1

Cancel the common factor.

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

Step 8.3.16.7.4.2

Rewrite the expression.

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

Step 8.3.16.7.5

Evaluate the exponent.

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

Step 8.3.17

Cancel the common factor of $\sqrt{3}$ .

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Step 8.3.17.1

Cancel the common factor.

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

Step 8.3.17.2

Rewrite the expression.

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

Step 8.3.18

Multiply $2$ by $3$ .

$\frac{1}{6} \cdot 3$

Step 8.3.19

Combine $\frac{1}{6}$ and $3$ .

$\frac{3}{6}$

Step 8.3.20

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

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Step 8.3.20.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{6}$

Step 8.3.20.2

Cancel the common factors.

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Step 8.3.20.2.1

Factor $3$ out of $6$ .

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

Step 8.3.20.2.2

Cancel the common factor.

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

Step 8.3.20.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 9

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

Center: $(- 1, 4)$

${Vertex}_{1}$ : $(- 1, 4 + \frac{\sqrt{3}}{3})$

${Vertex}_{2}$ : $(- 1, 4 - \frac{\sqrt{3}}{3})$

${Focus}_{1}$ : $(- 1, 4 + \frac{\sqrt{3}}{6})$

${Focus}_{2}$ : $(- 1, 4 - \frac{\sqrt{3}}{6})$

Eccentricity: $\frac{1}{2}$

Step 10