Precalculus Examples

$4 x^{2} + y^{2} + 8 x - 5 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Add $5$ to both sides of the equation.

$4 x^{2} + y^{2} + 8 x = 5$

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 $4 x^{2} + y^{2} + 8 x = 5$ .

$4 {(x + 1)}^{2} - 4 + y^{2} = 5$

Step 1.4

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

$4 {(x + 1)}^{2} + y^{2} = 5 + 4$

Step 1.5

Add $5$ and $4$ .

$4 {(x + 1)}^{2} + y^{2} = 9$

Step 1.6

Divide each term by $9$ to make the right side equal to one.

$\frac{4 {(x + 1)}^{2}}{9} + \frac{y^{2}}{9} = \frac{9}{9}$

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 $1$ .

$\frac{{(x + 1)}^{2}}{\frac{9}{4}} + \frac{y^{2}}{9} = 1$

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.

$a = 3$

$b = \frac{3}{2}$

$k = 0$

$h = - 1$

Step 4

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

$(- 1, 0)$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

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

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

Step 5.3.2

Apply the product rule to $\frac{3}{2}$ .

$\sqrt{9 - \frac{3^{2}}{2^{2}}}$

Step 5.3.3

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

$\sqrt{9 - \frac{9}{2^{2}}}$

Step 5.3.4

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

$\sqrt{9 - \frac{9}{4}}$

Step 5.3.5

To write $9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{9 \cdot \frac{4}{4} - \frac{9}{4}}$

Step 5.3.6

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

$\sqrt{\frac{9 \cdot 4}{4} - \frac{9}{4}}$

Step 5.3.7

Combine the numerators over the common denominator.

$\sqrt{\frac{9 \cdot 4 - 9}{4}}$

Step 5.3.8

Simplify the numerator.

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Step 5.3.8.1

Multiply $9$ by $4$ .

$\sqrt{\frac{36 - 9}{4}}$

Step 5.3.8.2

Subtract $9$ from $36$ .

$\sqrt{\frac{27}{4}}$

Step 5.3.9

Rewrite $\sqrt{\frac{27}{4}}$ as $\frac{\sqrt{27}}{\sqrt{4}}$ .

$\frac{\sqrt{27}}{\sqrt{4}}$

Step 5.3.10

Simplify the numerator.

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Step 5.3.10.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 5.3.10.1.1

Factor $9$ out of $27$ .

$\frac{\sqrt{9 (3)}}{\sqrt{4}}$

Step 5.3.10.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 3}}{\sqrt{4}}$

Step 5.3.10.2

Pull terms out from under the radical.

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

Step 5.3.11

Simplify the denominator.

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Step 5.3.11.1

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

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

Step 5.3.11.2

Pull terms out from under the radical, assuming positive real numbers.

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

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.

$(- 1, 0 + 3)$

Step 6.3

Simplify.

$(- 1, 3)$

Step 6.4

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

$(h, k - a)$

Step 6.5

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

$(- 1, 0 - (3))$

Step 6.6

Simplify.

$(- 1, - 3)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(- 1, 3)$

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

${Vertex}_{1}$ : $(- 1, 3)$

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

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.

$(- 1, 0 + \frac{3 \sqrt{3}}{2})$

Step 7.3

Simplify.

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

Step 7.4

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

$(h, k - c)$

Step 7.5

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

$(- 1, 0 - (\frac{3 \sqrt{3}}{2}))$

Step 7.6

Simplify.

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

Step 7.7

Ellipses have two foci.

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

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

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

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

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.

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

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

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

Step 8.3.1.2

Apply the product rule to $\frac{3}{2}$ .

$\frac{\sqrt{9 - \frac{3^{2}}{2^{2}}}}{3}$

Step 8.3.1.3

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

$\frac{\sqrt{9 - \frac{9}{2^{2}}}}{3}$

Step 8.3.1.4

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

$\frac{\sqrt{9 - \frac{9}{4}}}{3}$

Step 8.3.1.5

To write $9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\sqrt{9 \cdot \frac{4}{4} - \frac{9}{4}}}{3}$

Step 8.3.1.6

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

$\frac{\sqrt{\frac{9 \cdot 4}{4} - \frac{9}{4}}}{3}$

Step 8.3.1.7

Combine the numerators over the common denominator.

$\frac{\sqrt{\frac{9 \cdot 4 - 9}{4}}}{3}$

Step 8.3.1.8

Simplify the numerator.

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Step 8.3.1.8.1

Multiply $9$ by $4$ .

$\frac{\sqrt{\frac{36 - 9}{4}}}{3}$

Step 8.3.1.8.2

Subtract $9$ from $36$ .

$\frac{\sqrt{\frac{27}{4}}}{3}$

Step 8.3.1.9

Rewrite $\sqrt{\frac{27}{4}}$ as $\frac{\sqrt{27}}{\sqrt{4}}$ .

$\frac{\frac{\sqrt{27}}{\sqrt{4}}}{3}$

Step 8.3.1.10

Simplify the numerator.

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Step 8.3.1.10.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 8.3.1.10.1.1

Factor $9$ out of $27$ .

$\frac{\frac{\sqrt{9 (3)}}{\sqrt{4}}}{3}$

Step 8.3.1.10.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\frac{\sqrt{3^{2} \cdot 3}}{\sqrt{4}}}{3}$

Step 8.3.1.10.2

Pull terms out from under the radical.

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

Step 8.3.1.11

Simplify the denominator.

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Step 8.3.1.11.1

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

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

Step 8.3.1.11.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 8.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.3.3

Cancel the common factor of $3$ .

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Step 8.3.3.1

Factor $3$ out of $3 \sqrt{3}$ .

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

Step 8.3.3.2

Cancel the common factor.

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

Step 8.3.3.3

Rewrite the expression.

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

Step 9

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

Center: $(- 1, 0)$

${Vertex}_{1}$ : $(- 1, 3)$

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

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

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

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

Step 10