Precalculus Examples

$x^{2} + 5 y^{2} - 8 x - 30 y - 39 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Add $39$ to both sides of the equation.

$x^{2} + 5 y^{2} - 8 x - 30 y = 39$

Step 1.2

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

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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 $c$ .

$a = 1$

$b = - 8$

$c = 0$

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

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

Step 1.2.3.2

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

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Step 1.2.3.2.1

Factor $2$ out of $- 8$ .

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

Step 1.2.3.2.2

Cancel the common factors.

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Step 1.2.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 1.2.3.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

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

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 1}$

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 $2$ .

$e = 0 - \frac{64}{4 \cdot 1}$

Step 1.2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 1.2.4.2.1.3

Divide $64$ by $4$ .

$e = 0 - 1 \cdot 16$

Step 1.2.4.2.1.4

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Step 1.2.4.2.2

Subtract $16$ from $0$ .

$e = - 16$

Step 1.2.5

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

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

Step 1.3

Substitute ${(x - 4)}^{2} - 16$ for $x^{2} - 8 x$ in the equation $x^{2} + 5 y^{2} - 8 x - 30 y = 39$ .

${(x - 4)}^{2} - 16 + 5 y^{2} - 30 y = 39$

Step 1.4

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

${(x - 4)}^{2} + 5 y^{2} - 30 y = 39 + 16$

Step 1.5

Complete the square for $5 y^{2} - 30 y$ .

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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 $c$ .

$a = 5$

$b = - 30$

$c = 0$

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

$d = \frac{- 30}{2 \cdot 5}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $- 30$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $- 30$ .

$d = \frac{2 \cdot - 15}{2 \cdot 5}$

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 $2 \cdot 5$ .

$d = \frac{2 \cdot - 15}{2 (5)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 15}{2 \cdot 5}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 15}{5}$

Step 1.5.3.2.2

Cancel the common factor of $- 15$ and $5$ .

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Step 1.5.3.2.2.1

Factor $5$ out of $- 15$ .

$d = \frac{5 \cdot - 3}{5}$

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $5$ out of $5$ .

$d = \frac{5 \cdot - 3}{5 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{5 \cdot - 3}{5 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

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

Step 1.5.3.2.2.2.4

Divide $- 3$ by $1$ .

$d = - 3$

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

$e = 0 - \frac{{(- 30)}^{2}}{4 \cdot 5}$

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 $- 30$ to the power of $2$ .

$e = 0 - \frac{900}{4 \cdot 5}$

Step 1.5.4.2.1.2

Multiply $4$ by $5$ .

$e = 0 - \frac{900}{20}$

Step 1.5.4.2.1.3

Divide $900$ by $20$ .

$e = 0 - 1 \cdot 45$

Step 1.5.4.2.1.4

Multiply $- 1$ by $45$ .

$e = 0 - 45$

Step 1.5.4.2.2

Subtract $45$ from $0$ .

$e = - 45$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(y - 3)}^{2} - 45$ .

$5 {(y - 3)}^{2} - 45$

Step 1.6

Substitute $5 {(y - 3)}^{2} - 45$ for $5 y^{2} - 30 y$ in the equation $x^{2} + 5 y^{2} - 8 x - 30 y = 39$ .

${(x - 4)}^{2} + 5 {(y - 3)}^{2} - 45 = 39 + 16$

Step 1.7

Move $- 45$ to the right side of the equation by adding $45$ to both sides.

${(x - 4)}^{2} + 5 {(y - 3)}^{2} = 39 + 16 + 45$

Step 1.8

Simplify $39 + 16 + 45$ .

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Step 1.8.1

Add $39$ and $16$ .

${(x - 4)}^{2} + 5 {(y - 3)}^{2} = 55 + 45$

Step 1.8.2

Add $55$ and $45$ .

${(x - 4)}^{2} + 5 {(y - 3)}^{2} = 100$

Step 1.9

Divide each term by $100$ to make the right side equal to one.

$\frac{{(x - 4)}^{2}}{100} + \frac{5 {(y - 3)}^{2}}{100} = \frac{100}{100}$

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 $1$ .

$\frac{{(x - 4)}^{2}}{100} + \frac{{(y - 3)}^{2}}{20} = 1$

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.

$a = 10$

$b = 2 \sqrt{5}$

$k = 3$

$h = 4$

Step 4

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

$(4, 3)$

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.

$\sqrt{{(10)}^{2} - {(2 \sqrt{5})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the expression.

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Step 5.3.1.1

Raise $10$ to the power of $2$ .

$\sqrt{100 - {(2 \sqrt{5})}^{2}}$

Step 5.3.1.2

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

$\sqrt{100 - (2^{2} {\sqrt{5}}^{2})}$

Step 5.3.1.3

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

$\sqrt{100 - (4 {\sqrt{5}}^{2})}$

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 $5^{\frac{1}{2}}$ .

$\sqrt{100 - (4 {(5^{\frac{1}{2}})}^{2})}$

Step 5.3.2.2

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

$\sqrt{100 - (4 \cdot 5^{\frac{1}{2} \cdot 2})}$

Step 5.3.2.3

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

$\sqrt{100 - (4 \cdot 5^{\frac{2}{2}})}$

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.

$\sqrt{100 - (4 \cdot 5^{\frac{2}{2}})}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{100 - (4 \cdot 5^{1})}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{100 - (4 \cdot 5)}$

Step 5.3.3

Multiply $- (4 \cdot 5)$ .

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Step 5.3.3.1

Multiply $4$ by $5$ .

$\sqrt{100 - 1 \cdot 20}$

Step 5.3.3.2

Multiply $- 1$ by $20$ .

$\sqrt{100 - 20}$

Step 5.3.4

Subtract $20$ from $100$ .

$\sqrt{80}$

Step 5.3.5

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 5.3.5.1

Factor $16$ out of $80$ .

$\sqrt{16 (5)}$

Step 5.3.5.2

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \cdot 5}$

Step 5.3.6

Pull terms out from under the radical.

$4 \sqrt{5}$

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.

$(4 + 10, 3)$

Step 6.3

Simplify.

$(14, 3)$

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.

$(4 - (10), 3)$

Step 6.6

Simplify.

$(- 6, 3)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(14, 3)$

${Vertex}_{2}$ : $(- 6, 3)$

${Vertex}_{1}$ : $(14, 3)$

${Vertex}_{2}$ : $(- 6, 3)$

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 $h$ .

$(h + c, k)$

Step 7.2

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

$(4 + 4 \sqrt{5}, 3)$

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.

$(4 - (4 \sqrt{5}), 3)$

Step 7.5

Simplify.

$(4 - 4 \sqrt{5}, 3)$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(4 + 4 \sqrt{5}, 3)$

${Focus}_{2}$ : $(4 - 4 \sqrt{5}, 3)$

${Focus}_{1}$ : $(4 + 4 \sqrt{5}, 3)$

${Focus}_{2}$ : $(4 - 4 \sqrt{5}, 3)$

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.

$\frac{\sqrt{{(10)}^{2} - {(2 \sqrt{5})}^{2}}}{10}$

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 $10$ to the power of $2$ .

$\frac{\sqrt{100 - {(2 \sqrt{5})}^{2}}}{10}$

Step 8.3.1.2

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

$\frac{\sqrt{100 - (2^{2} {\sqrt{5}}^{2})}}{10}$

Step 8.3.1.3

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

$\frac{\sqrt{100 - (4 {\sqrt{5}}^{2})}}{10}$

Step 8.3.1.4

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

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Step 8.3.1.4.1

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

$\frac{\sqrt{100 - (4 {(5^{\frac{1}{2}})}^{2})}}{10}$

Step 8.3.1.4.2

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

$\frac{\sqrt{100 - (4 \cdot 5^{\frac{1}{2} \cdot 2})}}{10}$

Step 8.3.1.4.3

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

$\frac{\sqrt{100 - (4 \cdot 5^{\frac{2}{2}})}}{10}$

Step 8.3.1.4.4

Cancel the common factor of $2$ .

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Step 8.3.1.4.4.1

Cancel the common factor.

$\frac{\sqrt{100 - (4 \cdot 5^{\frac{2}{2}})}}{10}$

Step 8.3.1.4.4.2

Rewrite the expression.

$\frac{\sqrt{100 - (4 \cdot 5^{1})}}{10}$

Step 8.3.1.4.5

Evaluate the exponent.

$\frac{\sqrt{100 - (4 \cdot 5)}}{10}$

Step 8.3.1.5

Multiply $- (4 \cdot 5)$ .

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Step 8.3.1.5.1

Multiply $4$ by $5$ .

$\frac{\sqrt{100 - 1 \cdot 20}}{10}$

Step 8.3.1.5.2

Multiply $- 1$ by $20$ .

$\frac{\sqrt{100 - 20}}{10}$

Step 8.3.1.6

Subtract $20$ from $100$ .

$\frac{\sqrt{80}}{10}$

Step 8.3.1.7

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 8.3.1.7.1

Factor $16$ out of $80$ .

$\frac{\sqrt{16 (5)}}{10}$

Step 8.3.1.7.2

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2} \cdot 5}}{10}$

Step 8.3.1.8

Pull terms out from under the radical.

$\frac{4 \sqrt{5}}{10}$

Step 8.3.2

Cancel the common factor of $4$ and $10$ .

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Step 8.3.2.1

Factor $2$ out of $4 \sqrt{5}$ .

$\frac{2 (2 \sqrt{5})}{10}$

Step 8.3.2.2

Cancel the common factors.

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Step 8.3.2.2.1

Factor $2$ out of $10$ .

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

Step 8.3.2.2.2

Cancel the common factor.

$\frac{2 (2 \sqrt{5})}{2 \cdot 5}$

Step 8.3.2.2.3

Rewrite the expression.

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

Step 9

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

Center: $(4, 3)$

${Vertex}_{1}$ : $(14, 3)$

${Vertex}_{2}$ : $(- 6, 3)$

${Focus}_{1}$ : $(4 + 4 \sqrt{5}, 3)$

${Focus}_{2}$ : $(4 - 4 \sqrt{5}, 3)$

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

Step 10