Precalculus Examples

$3 x^{2} + 4 y^{2} + 12 x - 16 y - 32 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Add $32$ to both sides of the equation.

$3 x^{2} + 4 y^{2} + 12 x - 16 y = 32$

Step 1.2

Complete the square for $3 x^{2} + 12 x$ .

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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 $c$ .

$a = 3$

$b = 12$

$c = 0$

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 $3 {(x + 2)}^{2} - 12$ .

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

Step 1.3

Substitute $3 {(x + 2)}^{2} - 12$ for $3 x^{2} + 12 x$ in the equation $3 x^{2} + 4 y^{2} + 12 x - 16 y = 32$ .

$3 {(x + 2)}^{2} - 12 + 4 y^{2} - 16 y = 32$

Step 1.4

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

$3 {(x + 2)}^{2} + 4 y^{2} - 16 y = 32 + 12$

Step 1.5

Complete the square for $4 y^{2} - 16 y$ .

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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 $c$ .

$a = 4$

$b = - 16$

$c = 0$

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

$d = \frac{- 16}{2 \cdot 4}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $- 16$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $- 16$ .

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

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 $2 \cdot 4$ .

$d = \frac{2 \cdot - 8}{2 (4)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

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

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 8}{4}$

Step 1.5.3.2.2

Cancel the common factor of $- 8$ and $4$ .

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Step 1.5.3.2.2.1

Factor $4$ out of $- 8$ .

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

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $4$ out of $4$ .

$d = \frac{4 \cdot - 2}{4 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{4 \cdot - 2}{4 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

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

Step 1.5.3.2.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

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

$e = 0 - \frac{{(- 16)}^{2}}{4 \cdot 4}$

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 $- 16$ to the power of $2$ .

$e = 0 - \frac{256}{4 \cdot 4}$

Step 1.5.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{256}{16}$

Step 1.5.4.2.1.3

Divide $256$ by $16$ .

$e = 0 - 1 \cdot 16$

Step 1.5.4.2.1.4

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Step 1.5.4.2.2

Subtract $16$ from $0$ .

$e = - 16$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y - 2)}^{2} - 16$ .

$4 {(y - 2)}^{2} - 16$

Step 1.6

Substitute $4 {(y - 2)}^{2} - 16$ for $4 y^{2} - 16 y$ in the equation $3 x^{2} + 4 y^{2} + 12 x - 16 y = 32$ .

$3 {(x + 2)}^{2} + 4 {(y - 2)}^{2} - 16 = 32 + 12$

Step 1.7

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

$3 {(x + 2)}^{2} + 4 {(y - 2)}^{2} = 32 + 12 + 16$

Step 1.8

Simplify $32 + 12 + 16$ .

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Step 1.8.1

Add $32$ and $12$ .

$3 {(x + 2)}^{2} + 4 {(y - 2)}^{2} = 44 + 16$

Step 1.8.2

Add $44$ and $16$ .

$3 {(x + 2)}^{2} + 4 {(y - 2)}^{2} = 60$

Step 1.9

Divide each term by $60$ to make the right side equal to one.

$\frac{3 {(x + 2)}^{2}}{60} + \frac{4 {(y - 2)}^{2}}{60} = \frac{60}{60}$

Step 1.10

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 $1$ .

$\frac{{(x + 2)}^{2}}{20} + \frac{{(y - 2)}^{2}}{15} = 1$

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.

$a = 2 \sqrt{5}$

$b = \sqrt{15}$

$k = 2$

$h = - 2$

Step 4

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

$(- 2, 2)$

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.

$\sqrt{{(2 \sqrt{5})}^{2} - {(\sqrt{15})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the expression.

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Step 5.3.1.1

Apply the product rule to $2 \sqrt{5}$ .

$\sqrt{2^{2} {\sqrt{5}}^{2} - {(\sqrt{15})}^{2}}$

Step 5.3.1.2

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

$\sqrt{4 {\sqrt{5}}^{2} - {(\sqrt{15})}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{4 {(5^{\frac{1}{2}})}^{2} - {(\sqrt{15})}^{2}}$

Step 5.3.2.2

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

$\sqrt{4 \cdot 5^{\frac{1}{2} \cdot 2} - {(\sqrt{15})}^{2}}$

Step 5.3.2.3

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

$\sqrt{4 \cdot 5^{\frac{2}{2}} - {(\sqrt{15})}^{2}}$

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.

$\sqrt{4 \cdot 5^{\frac{2}{2}} - {(\sqrt{15})}^{2}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{4 \cdot 5^{1} - {(\sqrt{15})}^{2}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{4 \cdot 5 - {(\sqrt{15})}^{2}}$

Step 5.3.3

Multiply $4$ by $5$ .

$\sqrt{20 - {(\sqrt{15})}^{2}}$

Step 5.3.4

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 5.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\sqrt{20 - {(15^{\frac{1}{2}})}^{2}}$

Step 5.3.4.2

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

$\sqrt{20 - 15^{\frac{1}{2} \cdot 2}}$

Step 5.3.4.3

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

$\sqrt{20 - 15^{\frac{2}{2}}}$

Step 5.3.4.4

Cancel the common factor of $2$ .

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Step 5.3.4.4.1

Cancel the common factor.

$\sqrt{20 - 15^{\frac{2}{2}}}$

Step 5.3.4.4.2

Rewrite the expression.

$\sqrt{20 - 15^{1}}$

Step 5.3.4.5

Evaluate the exponent.

$\sqrt{20 - 1 \cdot 15}$

Step 5.3.5

Simplify the expression.

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Step 5.3.5.1

Multiply $- 1$ by $15$ .

$\sqrt{20 - 15}$

Step 5.3.5.2

Subtract $15$ from $20$ .

$\sqrt{5}$

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.

$(- 2 + 2 \sqrt{5}, 2)$

Step 6.3

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

$(h - a, k)$

Step 6.4

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

$(- 2 - (2 \sqrt{5}), 2)$

Step 6.5

Simplify.

$(- 2 - 2 \sqrt{5}, 2)$

Step 6.6

Ellipses have two vertices.

${Vertex}_{1}$ : $(- 2 + 2 \sqrt{5}, 2)$

${Vertex}_{2}$ : $(- 2 - 2 \sqrt{5}, 2)$

${Vertex}_{1}$ : $(- 2 + 2 \sqrt{5}, 2)$

${Vertex}_{2}$ : $(- 2 - 2 \sqrt{5}, 2)$

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.

$(- 2 + \sqrt{5}, 2)$

Step 7.3

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

$(h - c, k)$

Step 7.4

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

$(- 2 - (\sqrt{5}), 2)$

Step 7.5

Simplify.

$(- 2 - \sqrt{5}, 2)$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(- 2 + \sqrt{5}, 2)$

${Focus}_{2}$ : $(- 2 - \sqrt{5}, 2)$

${Focus}_{1}$ : $(- 2 + \sqrt{5}, 2)$

${Focus}_{2}$ : $(- 2 - \sqrt{5}, 2)$

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.

$\frac{\sqrt{{(2 \sqrt{5})}^{2} - {(\sqrt{15})}^{2}}}{2 \sqrt{5}}$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Apply the product rule to $2 \sqrt{5}$ .

$\frac{\sqrt{2^{2} {\sqrt{5}}^{2} - {\sqrt{15}}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.2

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

$\frac{\sqrt{4 {\sqrt{5}}^{2} - {\sqrt{15}}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.3

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.3.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{4 {(5^{\frac{1}{2}})}^{2} - {\sqrt{15}}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.3.2

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

$\frac{\sqrt{4 \cdot 5^{\frac{1}{2} \cdot 2} - {\sqrt{15}}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.3.3

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

$\frac{\sqrt{4 \cdot 5^{\frac{2}{2}} - {\sqrt{15}}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.3.4

Cancel the common factor of $2$ .

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Step 8.3.1.3.4.1

Cancel the common factor.

$\frac{\sqrt{4 \cdot 5^{\frac{2}{2}} - {\sqrt{15}}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.3.4.2

Rewrite the expression.

$\frac{\sqrt{4 \cdot 5^{1} - {\sqrt{15}}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.3.5

Evaluate the exponent.

$\frac{\sqrt{4 \cdot 5 - {\sqrt{15}}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.4

Multiply $4$ by $5$ .

$\frac{\sqrt{20 - {\sqrt{15}}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.5

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 8.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{\sqrt{20 - {(15^{\frac{1}{2}})}^{2}}}{2 \sqrt{5}}$

Step 8.3.1.5.2

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

$\frac{\sqrt{20 - 15^{\frac{1}{2} \cdot 2}}}{2 \sqrt{5}}$

Step 8.3.1.5.3

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

$\frac{\sqrt{20 - 15^{\frac{2}{2}}}}{2 \sqrt{5}}$

Step 8.3.1.5.4

Cancel the common factor of $2$ .

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Step 8.3.1.5.4.1

Cancel the common factor.

$\frac{\sqrt{20 - 15^{\frac{2}{2}}}}{2 \sqrt{5}}$

Step 8.3.1.5.4.2

Rewrite the expression.

$\frac{\sqrt{20 - 15^{1}}}{2 \sqrt{5}}$

Step 8.3.1.5.5

Evaluate the exponent.

$\frac{\sqrt{20 - 1 \cdot 15}}{2 \sqrt{5}}$

Step 8.3.1.6

Multiply $- 1$ by $15$ .

$\frac{\sqrt{20 - 15}}{2 \sqrt{5}}$

Step 8.3.1.7

Subtract $15$ from $20$ .

$\frac{\sqrt{5}}{2 \sqrt{5}}$

Step 8.3.2

Cancel the common factor of $\sqrt{5}$ .

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Step 8.3.2.1

Cancel the common factor.

$\frac{\sqrt{5}}{2 \sqrt{5}}$

Step 8.3.2.2

Rewrite the expression.

$\frac{1}{2}$

Step 9

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

Center: $(- 2, 2)$

${Vertex}_{1}$ : $(- 2 + 2 \sqrt{5}, 2)$

${Vertex}_{2}$ : $(- 2 - 2 \sqrt{5}, 2)$

${Focus}_{1}$ : $(- 2 + \sqrt{5}, 2)$

${Focus}_{2}$ : $(- 2 - \sqrt{5}, 2)$

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

Step 10