Precalculus Examples

$8 x^{2} + 6 y^{2} - 32 x + 24 y + 8 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Subtract $8$ from both sides of the equation.

$8 x^{2} + 6 y^{2} - 32 x + 24 y = - 8$

Step 1.2

Complete the square for $8 x^{2} - 32 x$ .

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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 $c$ .

$a = 8$

$b = - 32$

$c = 0$

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

$d = \frac{- 32}{2 \cdot 8}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 32$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 32$ .

$d = \frac{2 \cdot - 16}{2 \cdot 8}$

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 $2 \cdot 8$ .

$d = \frac{2 \cdot - 16}{2 (8)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 16}{2 \cdot 8}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 16}{8}$

Step 1.2.3.2.2

Cancel the common factor of $- 16$ and $8$ .

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Step 1.2.3.2.2.1

Factor $8$ out of $- 16$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $8$ out of $8$ .

$d = \frac{8 \cdot - 2}{8 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{8 \cdot - 2}{8 \cdot 1}$

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

$e = 0 - \frac{{(- 32)}^{2}}{4 \cdot 8}$

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 $- 32$ to the power of $2$ .

$e = 0 - \frac{1024}{4 \cdot 8}$

Step 1.2.4.2.1.2

Multiply $4$ by $8$ .

$e = 0 - \frac{1024}{32}$

Step 1.2.4.2.1.3

Divide $1024$ by $32$ .

$e = 0 - 1 \cdot 32$

Step 1.2.4.2.1.4

Multiply $- 1$ by $32$ .

$e = 0 - 32$

Step 1.2.4.2.2

Subtract $32$ from $0$ .

$e = - 32$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $8 {(x - 2)}^{2} - 32$ .

$8 {(x - 2)}^{2} - 32$

Step 1.3

Substitute $8 {(x - 2)}^{2} - 32$ for $8 x^{2} - 32 x$ in the equation $8 x^{2} + 6 y^{2} - 32 x + 24 y = - 8$ .

$8 {(x - 2)}^{2} - 32 + 6 y^{2} + 24 y = - 8$

Step 1.4

Move $- 32$ to the right side of the equation by adding $32$ to both sides.

$8 {(x - 2)}^{2} + 6 y^{2} + 24 y = - 8 + 32$

Step 1.5

Complete the square for $6 y^{2} + 24 y$ .

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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 $c$ .

$a = 6$

$b = 24$

$c = 0$

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

$d = \frac{24}{2 \cdot 6}$

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 $24$ .

$d = \frac{2 \cdot 12}{2 \cdot 6}$

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 $2 \cdot 6$ .

$d = \frac{2 \cdot 12}{2 (6)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 12}{2 \cdot 6}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{12}{6}$

Step 1.5.3.2.2

Cancel the common factor of $12$ and $6$ .

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Step 1.5.3.2.2.1

Factor $6$ out of $12$ .

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

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $6$ out of $6$ .

$d = \frac{6 \cdot 2}{6 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{6 \cdot 2}{6 \cdot 1}$

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

$e = 0 - \frac{24^{2}}{4 \cdot 6}$

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 $2$ .

$e = 0 - \frac{576}{4 \cdot 6}$

Step 1.5.4.2.1.2

Multiply $4$ by $6$ .

$e = 0 - \frac{576}{24}$

Step 1.5.4.2.1.3

Divide $576$ by $24$ .

$e = 0 - 1 \cdot 24$

Step 1.5.4.2.1.4

Multiply $- 1$ by $24$ .

$e = 0 - 24$

Step 1.5.4.2.2

Subtract $24$ from $0$ .

$e = - 24$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $6 {(y + 2)}^{2} - 24$ .

$6 {(y + 2)}^{2} - 24$

Step 1.6

Substitute $6 {(y + 2)}^{2} - 24$ for $6 y^{2} + 24 y$ in the equation $8 x^{2} + 6 y^{2} - 32 x + 24 y = - 8$ .

$8 {(x - 2)}^{2} + 6 {(y + 2)}^{2} - 24 = - 8 + 32$

Step 1.7

Move $- 24$ to the right side of the equation by adding $24$ to both sides.

$8 {(x - 2)}^{2} + 6 {(y + 2)}^{2} = - 8 + 32 + 24$

Step 1.8

Simplify $- 8 + 32 + 24$ .

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Step 1.8.1

Add $- 8$ and $32$ .

$8 {(x - 2)}^{2} + 6 {(y + 2)}^{2} = 24 + 24$

Step 1.8.2

Add $24$ and $24$ .

$8 {(x - 2)}^{2} + 6 {(y + 2)}^{2} = 48$

Step 1.9

Divide each term by $48$ to make the right side equal to one.

$\frac{8 {(x - 2)}^{2}}{48} + \frac{6 {(y + 2)}^{2}}{48} = \frac{48}{48}$

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 $1$ .

$\frac{{(x - 2)}^{2}}{6} + \frac{{(y + 2)}^{2}}{8} = 1$

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.

$a = 2 \sqrt{2}$

$b = \sqrt{6}$

$k = - 2$

$h = 2$

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.

$\sqrt{{(2 \sqrt{2})}^{2} - {(\sqrt{6})}^{2}}$

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 $2 \sqrt{2}$ .

$\sqrt{2^{2} {\sqrt{2}}^{2} - {(\sqrt{6})}^{2}}$

Step 5.3.1.2

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

$\sqrt{4 {\sqrt{2}}^{2} - {(\sqrt{6})}^{2}}$

Step 5.3.2

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

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Step 5.3.2.1

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

$\sqrt{4 {(2^{\frac{1}{2}})}^{2} - {(\sqrt{6})}^{2}}$

Step 5.3.2.2

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

$\sqrt{4 \cdot 2^{\frac{1}{2} \cdot 2} - {(\sqrt{6})}^{2}}$

Step 5.3.2.3

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

$\sqrt{4 \cdot 2^{\frac{2}{2}} - {(\sqrt{6})}^{2}}$

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.

$\sqrt{4 \cdot 2^{\frac{2}{2}} - {(\sqrt{6})}^{2}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{4 \cdot 2^{1} - {(\sqrt{6})}^{2}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{4 \cdot 2 - {(\sqrt{6})}^{2}}$

Step 5.3.3

Multiply $4$ by $2$ .

$\sqrt{8 - {(\sqrt{6})}^{2}}$

Step 5.3.4

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 5.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{8 - {(6^{\frac{1}{2}})}^{2}}$

Step 5.3.4.2

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

$\sqrt{8 - 6^{\frac{1}{2} \cdot 2}}$

Step 5.3.4.3

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

$\sqrt{8 - 6^{\frac{2}{2}}}$

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.

$\sqrt{8 - 6^{\frac{2}{2}}}$

Step 5.3.4.4.2

Rewrite the expression.

$\sqrt{8 - 6^{1}}$

Step 5.3.4.5

Evaluate the exponent.

$\sqrt{8 - 1 \cdot 6}$

Step 5.3.5

Simplify the expression.

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Step 5.3.5.1

Multiply $- 1$ by $6$ .

$\sqrt{8 - 6}$

Step 5.3.5.2

Subtract $6$ from $8$ .

$\sqrt{2}$

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 $k$ .

$(h, k + a)$

Step 6.2

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

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

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.

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

Step 6.5

Simplify.

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

Step 6.6

Ellipses have two vertices.

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

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

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

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

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 $k$ .

$(h, k + c)$

Step 7.2

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

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

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.

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

Step 7.5

Simplify.

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

Step 7.6

Ellipses have two foci.

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

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

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

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

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.

$\frac{\sqrt{{(2 \sqrt{2})}^{2} - {(\sqrt{6})}^{2}}}{2 \sqrt{2}}$

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 $2 \sqrt{2}$ .

$\frac{\sqrt{2^{2} {\sqrt{2}}^{2} - {\sqrt{6}}^{2}}}{2 \sqrt{2}}$

Step 8.3.1.2

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

$\frac{\sqrt{4 {\sqrt{2}}^{2} - {\sqrt{6}}^{2}}}{2 \sqrt{2}}$

Step 8.3.1.3

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

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Step 8.3.1.3.1

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

$\frac{\sqrt{4 {(2^{\frac{1}{2}})}^{2} - {\sqrt{6}}^{2}}}{2 \sqrt{2}}$

Step 8.3.1.3.2

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

$\frac{\sqrt{4 \cdot 2^{\frac{1}{2} \cdot 2} - {\sqrt{6}}^{2}}}{2 \sqrt{2}}$

Step 8.3.1.3.3

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

$\frac{\sqrt{4 \cdot 2^{\frac{2}{2}} - {\sqrt{6}}^{2}}}{2 \sqrt{2}}$

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.

$\frac{\sqrt{4 \cdot 2^{\frac{2}{2}} - {\sqrt{6}}^{2}}}{2 \sqrt{2}}$

Step 8.3.1.3.4.2

Rewrite the expression.

$\frac{\sqrt{4 \cdot 2^{1} - {\sqrt{6}}^{2}}}{2 \sqrt{2}}$

Step 8.3.1.3.5

Evaluate the exponent.

$\frac{\sqrt{4 \cdot 2 - {\sqrt{6}}^{2}}}{2 \sqrt{2}}$

Step 8.3.1.4

Multiply $4$ by $2$ .

$\frac{\sqrt{8 - {\sqrt{6}}^{2}}}{2 \sqrt{2}}$

Step 8.3.1.5

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 8.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{8 - {(6^{\frac{1}{2}})}^{2}}}{2 \sqrt{2}}$

Step 8.3.1.5.2

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

$\frac{\sqrt{8 - 6^{\frac{1}{2} \cdot 2}}}{2 \sqrt{2}}$

Step 8.3.1.5.3

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

$\frac{\sqrt{8 - 6^{\frac{2}{2}}}}{2 \sqrt{2}}$

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.

$\frac{\sqrt{8 - 6^{\frac{2}{2}}}}{2 \sqrt{2}}$

Step 8.3.1.5.4.2

Rewrite the expression.

$\frac{\sqrt{8 - 6^{1}}}{2 \sqrt{2}}$

Step 8.3.1.5.5

Evaluate the exponent.

$\frac{\sqrt{8 - 1 \cdot 6}}{2 \sqrt{2}}$

Step 8.3.1.6

Multiply $- 1$ by $6$ .

$\frac{\sqrt{8 - 6}}{2 \sqrt{2}}$

Step 8.3.1.7

Subtract $6$ from $8$ .

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

Step 8.3.2

Cancel the common factor of $\sqrt{2}$ .

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Step 8.3.2.1

Cancel the common factor.

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

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: $(2, - 2)$

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

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

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

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

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

Step 10