Pre-Algebra Examples

$5 x^{2} + 2 y^{2} - 27 x + 16 y - 19 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Add $19$ to both sides of the equation.

$5 x^{2} + 2 y^{2} - 27 x + 16 y = 19$

Step 1.2

Complete the square for $5 x^{2} - 27 x$ .

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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 $c$ .

$a = 5$

$b = - 27$

$c = 0$

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

$d = \frac{- 27}{2 \cdot 5}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $2$ by $5$ .

$d = \frac{- 27}{10}$

Step 1.2.3.2.2

Move the negative in front of the fraction.

$d = - \frac{27}{10}$

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

$e = 0 - \frac{{(- 27)}^{2}}{4 \cdot 5}$

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 $- 27$ to the power of $2$ .

$e = 0 - \frac{729}{4 \cdot 5}$

Step 1.2.4.2.1.2

Multiply $4$ by $5$ .

$e = 0 - \frac{729}{20}$

Step 1.2.4.2.2

Subtract $\frac{729}{20}$ from $0$ .

$e = - \frac{729}{20}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x - \frac{27}{10})}^{2} - \frac{729}{20}$ .

$5 {(x - \frac{27}{10})}^{2} - \frac{729}{20}$

Step 1.3

Substitute $5 {(x - \frac{27}{10})}^{2} - \frac{729}{20}$ for $5 x^{2} - 27 x$ in the equation $5 x^{2} + 2 y^{2} - 27 x + 16 y = 19$ .

$5 {(x - \frac{27}{10})}^{2} - \frac{729}{20} + 2 y^{2} + 16 y = 19$

Step 1.4

Move $- \frac{729}{20}$ to the right side of the equation by adding $\frac{729}{20}$ to both sides.

$5 {(x - \frac{27}{10})}^{2} + 2 y^{2} + 16 y = 19 + \frac{729}{20}$

Step 1.5

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

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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 $c$ .

$a = 2$

$b = 16$

$c = 0$

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

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

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 $16$ .

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

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 $2 \cdot 2$ .

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

Step 1.5.3.2.1.2.2

Cancel the common factor.

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

Step 1.5.3.2.1.2.3

Rewrite the expression.

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

Step 1.5.3.2.2

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

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Step 1.5.3.2.2.1

Factor $2$ out of $8$ .

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

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 1.5.3.2.2.2.2

Cancel the common factor.

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

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

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

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 $2$ .

$e = 0 - \frac{256}{4 \cdot 2}$

Step 1.5.4.2.1.2

Multiply $4$ by $2$ .

$e = 0 - \frac{256}{8}$

Step 1.5.4.2.1.3

Divide $256$ by $8$ .

$e = 0 - 1 \cdot 32$

Step 1.5.4.2.1.4

Multiply $- 1$ by $32$ .

$e = 0 - 32$

Step 1.5.4.2.2

Subtract $32$ from $0$ .

$e = - 32$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(y + 4)}^{2} - 32$ .

$2 {(y + 4)}^{2} - 32$

Step 1.6

Substitute $2 {(y + 4)}^{2} - 32$ for $2 y^{2} + 16 y$ in the equation $5 x^{2} + 2 y^{2} - 27 x + 16 y = 19$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} - 32 = 19 + \frac{729}{20}$

Step 1.7

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

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = 19 + \frac{729}{20} + 32$

Step 1.8

Simplify $19 + \frac{729}{20} + 32$ .

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Step 1.8.1

Find the common denominator.

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Step 1.8.1.1

Write $19$ as a fraction with denominator $1$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{19}{1} + \frac{729}{20} + 32$

Step 1.8.1.2

Multiply $\frac{19}{1}$ by $\frac{20}{20}$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{19}{1} \cdot \frac{20}{20} + \frac{729}{20} + 32$

Step 1.8.1.3

Multiply $\frac{19}{1}$ by $\frac{20}{20}$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{19 \cdot 20}{20} + \frac{729}{20} + 32$

Step 1.8.1.4

Write $32$ as a fraction with denominator $1$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{19 \cdot 20}{20} + \frac{729}{20} + \frac{32}{1}$

Step 1.8.1.5

Multiply $\frac{32}{1}$ by $\frac{20}{20}$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{19 \cdot 20}{20} + \frac{729}{20} + \frac{32}{1} \cdot \frac{20}{20}$

Step 1.8.1.6

Multiply $\frac{32}{1}$ by $\frac{20}{20}$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{19 \cdot 20}{20} + \frac{729}{20} + \frac{32 \cdot 20}{20}$

Step 1.8.2

Combine the numerators over the common denominator.

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{19 \cdot 20 + 729 + 32 \cdot 20}{20}$

Step 1.8.3

Simplify each term.

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Step 1.8.3.1

Multiply $19$ by $20$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{380 + 729 + 32 \cdot 20}{20}$

Step 1.8.3.2

Multiply $32$ by $20$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{380 + 729 + 640}{20}$

Step 1.8.4

Simplify by adding numbers.

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Step 1.8.4.1

Add $380$ and $729$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{1109 + 640}{20}$

Step 1.8.4.2

Add $1109$ and $640$ .

$5 {(x - \frac{27}{10})}^{2} + 2 {(y + 4)}^{2} = \frac{1749}{20}$

Step 1.9

Divide each term by $\frac{1749}{20}$ to make the right side equal to one.

$\frac{5 {(x - \frac{27}{10})}^{2}}{\frac{1749}{20}} + \frac{2 {(y + 4)}^{2}}{\frac{1749}{20}} = \frac{\frac{1749}{20}}{\frac{1749}{20}}$

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 $1$ .

$\frac{{(x - \frac{27}{10})}^{2}}{\frac{1749}{100}} + \frac{{(y + 4)}^{2}}{\frac{1749}{40}} = 1$

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.

$a = \frac{\sqrt{17490}}{20}$

$b = \frac{\sqrt{1749}}{10}$

$k = - 4$

$h = \frac{27}{10}$

Step 4

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

$(\frac{27}{10}, - 4)$

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.

$\sqrt{{(\frac{\sqrt{17490}}{20})}^{2} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Apply the product rule to $\frac{\sqrt{17490}}{20}$ .

$\sqrt{\frac{{\sqrt{17490}}^{2}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{17490}}^{2}$ as $17490$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17490}$ as $17490^{\frac{1}{2}}$ .

$\sqrt{\frac{{(17490^{\frac{1}{2}})}^{2}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.2.2

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

$\sqrt{\frac{17490^{\frac{1}{2} \cdot 2}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.2.3

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

$\sqrt{\frac{17490^{\frac{2}{2}}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}}$

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.

$\sqrt{\frac{17490^{\frac{2}{2}}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{\frac{17490^{1}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{\frac{17490}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.3

Raise $20$ to the power of $2$ .

$\sqrt{\frac{17490}{400} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.4

Cancel the common factor of $17490$ and $400$ .

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Step 5.3.4.1

Factor $10$ out of $17490$ .

$\sqrt{\frac{10 (1749)}{400} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.4.2

Cancel the common factors.

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Step 5.3.4.2.1

Factor $10$ out of $400$ .

$\sqrt{\frac{10 \cdot 1749}{10 \cdot 40} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.4.2.2

Cancel the common factor.

$\sqrt{\frac{10 \cdot 1749}{10 \cdot 40} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.4.2.3

Rewrite the expression.

$\sqrt{\frac{1749}{40} - {(\frac{\sqrt{1749}}{10})}^{2}}$

Step 5.3.5

Apply the product rule to $\frac{\sqrt{1749}}{10}$ .

$\sqrt{\frac{1749}{40} - \frac{{\sqrt{1749}}^{2}}{10^{2}}}$

Step 5.3.6

Rewrite ${\sqrt{1749}}^{2}$ as $1749$ .

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Step 5.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1749}$ as $1749^{\frac{1}{2}}$ .

$\sqrt{\frac{1749}{40} - \frac{{(1749^{\frac{1}{2}})}^{2}}{10^{2}}}$

Step 5.3.6.2

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

$\sqrt{\frac{1749}{40} - \frac{1749^{\frac{1}{2} \cdot 2}}{10^{2}}}$

Step 5.3.6.3

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

$\sqrt{\frac{1749}{40} - \frac{1749^{\frac{2}{2}}}{10^{2}}}$

Step 5.3.6.4

Cancel the common factor of $2$ .

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Step 5.3.6.4.1

Cancel the common factor.

$\sqrt{\frac{1749}{40} - \frac{1749^{\frac{2}{2}}}{10^{2}}}$

Step 5.3.6.4.2

Rewrite the expression.

$\sqrt{\frac{1749}{40} - \frac{1749^{1}}{10^{2}}}$

Step 5.3.6.5

Evaluate the exponent.

$\sqrt{\frac{1749}{40} - \frac{1749}{10^{2}}}$

Step 5.3.7

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

$\sqrt{\frac{1749}{40} - \frac{1749}{100}}$

Step 5.3.8

To write $\frac{1749}{40}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\sqrt{\frac{1749}{40} \cdot \frac{5}{5} - \frac{1749}{100}}$

Step 5.3.9

To write $- \frac{1749}{100}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{1749}{40} \cdot \frac{5}{5} - \frac{1749}{100} \cdot \frac{2}{2}}$

Step 5.3.10

Write each expression with a common denominator of $200$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.10.1

Multiply $\frac{1749}{40}$ by $\frac{5}{5}$ .

$\sqrt{\frac{1749 \cdot 5}{40 \cdot 5} - \frac{1749}{100} \cdot \frac{2}{2}}$

Step 5.3.10.2

Multiply $40$ by $5$ .

$\sqrt{\frac{1749 \cdot 5}{200} - \frac{1749}{100} \cdot \frac{2}{2}}$

Step 5.3.10.3

Multiply $\frac{1749}{100}$ by $\frac{2}{2}$ .

$\sqrt{\frac{1749 \cdot 5}{200} - \frac{1749 \cdot 2}{100 \cdot 2}}$

Step 5.3.10.4

Multiply $100$ by $2$ .

$\sqrt{\frac{1749 \cdot 5}{200} - \frac{1749 \cdot 2}{200}}$

Step 5.3.11

Combine the numerators over the common denominator.

$\sqrt{\frac{1749 \cdot 5 - 1749 \cdot 2}{200}}$

Step 5.3.12

Simplify the numerator.

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Step 5.3.12.1

Multiply $1749$ by $5$ .

$\sqrt{\frac{8745 - 1749 \cdot 2}{200}}$

Step 5.3.12.2

Multiply $- 1749$ by $2$ .

$\sqrt{\frac{8745 - 3498}{200}}$

Step 5.3.12.3

Subtract $3498$ from $8745$ .

$\sqrt{\frac{5247}{200}}$

Step 5.3.13

Rewrite $\sqrt{\frac{5247}{200}}$ as $\frac{\sqrt{5247}}{\sqrt{200}}$ .

$\frac{\sqrt{5247}}{\sqrt{200}}$

Step 5.3.14

Simplify the numerator.

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Step 5.3.14.1

Rewrite $5247$ as $3^{2} \cdot 583$ .

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Step 5.3.14.1.1

Factor $9$ out of $5247$ .

$\frac{\sqrt{9 (583)}}{\sqrt{200}}$

Step 5.3.14.1.2

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

$\frac{\sqrt{3^{2} \cdot 583}}{\sqrt{200}}$

Step 5.3.14.2

Pull terms out from under the radical.

$\frac{3 \sqrt{583}}{\sqrt{200}}$

Step 5.3.15

Simplify the denominator.

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Step 5.3.15.1

Rewrite $200$ as $10^{2} \cdot 2$ .

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Step 5.3.15.1.1

Factor $100$ out of $200$ .

$\frac{3 \sqrt{583}}{\sqrt{100 (2)}}$

Step 5.3.15.1.2

Rewrite $100$ as $10^{2}$ .

$\frac{3 \sqrt{583}}{\sqrt{10^{2} \cdot 2}}$

Step 5.3.15.2

Pull terms out from under the radical.

$\frac{3 \sqrt{583}}{10 \sqrt{2}}$

Step 5.3.16

Multiply $\frac{3 \sqrt{583}}{10 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{583}}{10 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.3.17

Combine and simplify the denominator.

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Step 5.3.17.1

Multiply $\frac{3 \sqrt{583}}{10 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{583} \sqrt{2}}{10 \sqrt{2} \sqrt{2}}$

Step 5.3.17.2

Move $\sqrt{2}$ .

$\frac{3 \sqrt{583} \sqrt{2}}{10 (\sqrt{2} \sqrt{2})}$

Step 5.3.17.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 \sqrt{583} \sqrt{2}}{10 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 5.3.17.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 \sqrt{583} \sqrt{2}}{10 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 5.3.17.5

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

$\frac{3 \sqrt{583} \sqrt{2}}{10 {\sqrt{2}}^{1 + 1}}$

Step 5.3.17.6

Add $1$ and $1$ .

$\frac{3 \sqrt{583} \sqrt{2}}{10 {\sqrt{2}}^{2}}$

Step 5.3.17.7

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

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Step 5.3.17.7.1

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

$\frac{3 \sqrt{583} \sqrt{2}}{10 {(2^{\frac{1}{2}})}^{2}}$

Step 5.3.17.7.2

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

$\frac{3 \sqrt{583} \sqrt{2}}{10 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 5.3.17.7.3

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

$\frac{3 \sqrt{583} \sqrt{2}}{10 \cdot 2^{\frac{2}{2}}}$

Step 5.3.17.7.4

Cancel the common factor of $2$ .

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Step 5.3.17.7.4.1

Cancel the common factor.

$\frac{3 \sqrt{583} \sqrt{2}}{10 \cdot 2^{\frac{2}{2}}}$

Step 5.3.17.7.4.2

Rewrite the expression.

$\frac{3 \sqrt{583} \sqrt{2}}{10 \cdot 2^{1}}$

Step 5.3.17.7.5

Evaluate the exponent.

$\frac{3 \sqrt{583} \sqrt{2}}{10 \cdot 2}$

Step 5.3.18

Simplify the numerator.

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Step 5.3.18.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{2 \cdot 583}}{10 \cdot 2}$

Step 5.3.18.2

Multiply $2$ by $583$ .

$\frac{3 \sqrt{1166}}{10 \cdot 2}$

Step 5.3.19

Multiply $10$ by $2$ .

$\frac{3 \sqrt{1166}}{20}$

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.

$(\frac{27}{10}, - 4 + \frac{\sqrt{17490}}{20})$

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.

$(\frac{27}{10}, - 4 - (\frac{\sqrt{17490}}{20}))$

Step 6.5

Simplify.

$(\frac{27}{10}, - 4 - \frac{\sqrt{17490}}{20})$

Step 6.6

Ellipses have two vertices.

${Vertex}_{1}$ : $(\frac{27}{10}, - 4 + \frac{\sqrt{17490}}{20})$

${Vertex}_{2}$ : $(\frac{27}{10}, - 4 - \frac{\sqrt{17490}}{20})$

${Vertex}_{1}$ : $(\frac{27}{10}, - 4 + \frac{\sqrt{17490}}{20})$

${Vertex}_{2}$ : $(\frac{27}{10}, - 4 - \frac{\sqrt{17490}}{20})$

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.

$(\frac{27}{10}, - 4 + \frac{3 \sqrt{1166}}{20})$

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.

$(\frac{27}{10}, - 4 - (\frac{3 \sqrt{1166}}{20}))$

Step 7.5

Simplify.

$(\frac{27}{10}, - 4 - \frac{3 \sqrt{1166}}{20})$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(\frac{27}{10}, - 4 + \frac{3 \sqrt{1166}}{20})$

${Focus}_{2}$ : $(\frac{27}{10}, - 4 - \frac{3 \sqrt{1166}}{20})$

${Focus}_{1}$ : $(\frac{27}{10}, - 4 + \frac{3 \sqrt{1166}}{20})$

${Focus}_{2}$ : $(\frac{27}{10}, - 4 - \frac{3 \sqrt{1166}}{20})$

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.

$\frac{\sqrt{{(\frac{\sqrt{17490}}{20})}^{2} - {(\frac{\sqrt{1749}}{10})}^{2}}}{\frac{\sqrt{17490}}{20}}$

Step 8.3

Simplify.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{{(\frac{\sqrt{17490}}{20})}^{2} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.2

Apply the product rule to $\frac{\sqrt{17490}}{20}$ .

$\sqrt{\frac{{\sqrt{17490}}^{2}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.3

Rewrite ${\sqrt{17490}}^{2}$ as $17490$ .

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Step 8.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17490}$ as $17490^{\frac{1}{2}}$ .

$\sqrt{\frac{{(17490^{\frac{1}{2}})}^{2}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.3.2

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

$\sqrt{\frac{17490^{\frac{1}{2} \cdot 2}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.3.3

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

$\sqrt{\frac{17490^{\frac{2}{2}}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

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.

$\sqrt{\frac{17490^{\frac{2}{2}}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.3.4.2

Rewrite the expression.

$\sqrt{\frac{17490^{1}}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.3.5

Evaluate the exponent.

$\sqrt{\frac{17490}{20^{2}} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.4

Raise $20$ to the power of $2$ .

$\sqrt{\frac{17490}{400} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.5

Cancel the common factor of $17490$ and $400$ .

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Step 8.3.5.1

Factor $10$ out of $17490$ .

$\sqrt{\frac{10 (1749)}{400} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.5.2

Cancel the common factors.

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Step 8.3.5.2.1

Factor $10$ out of $400$ .

$\sqrt{\frac{10 \cdot 1749}{10 \cdot 40} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.5.2.2

Cancel the common factor.

$\sqrt{\frac{10 \cdot 1749}{10 \cdot 40} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.5.2.3

Rewrite the expression.

$\sqrt{\frac{1749}{40} - {(\frac{\sqrt{1749}}{10})}^{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.6

Apply the product rule to $\frac{\sqrt{1749}}{10}$ .

$\sqrt{\frac{1749}{40} - \frac{{\sqrt{1749}}^{2}}{10^{2}}} \frac{20}{\sqrt{17490}}$

Step 8.3.7

Rewrite ${\sqrt{1749}}^{2}$ as $1749$ .

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Step 8.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1749}$ as $1749^{\frac{1}{2}}$ .

$\sqrt{\frac{1749}{40} - \frac{{(1749^{\frac{1}{2}})}^{2}}{10^{2}}} \frac{20}{\sqrt{17490}}$

Step 8.3.7.2

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

$\sqrt{\frac{1749}{40} - \frac{1749^{\frac{1}{2} \cdot 2}}{10^{2}}} \frac{20}{\sqrt{17490}}$

Step 8.3.7.3

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

$\sqrt{\frac{1749}{40} - \frac{1749^{\frac{2}{2}}}{10^{2}}} \frac{20}{\sqrt{17490}}$

Step 8.3.7.4

Cancel the common factor of $2$ .

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Step 8.3.7.4.1

Cancel the common factor.

$\sqrt{\frac{1749}{40} - \frac{1749^{\frac{2}{2}}}{10^{2}}} \frac{20}{\sqrt{17490}}$

Step 8.3.7.4.2

Rewrite the expression.

$\sqrt{\frac{1749}{40} - \frac{1749^{1}}{10^{2}}} \frac{20}{\sqrt{17490}}$

Step 8.3.7.5

Evaluate the exponent.

$\sqrt{\frac{1749}{40} - \frac{1749}{10^{2}}} \frac{20}{\sqrt{17490}}$

Step 8.3.8

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

$\sqrt{\frac{1749}{40} - \frac{1749}{100}} \frac{20}{\sqrt{17490}}$

Step 8.3.9

To write $\frac{1749}{40}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\sqrt{\frac{1749}{40} \cdot \frac{5}{5} - \frac{1749}{100}} \frac{20}{\sqrt{17490}}$

Step 8.3.10

To write $- \frac{1749}{100}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{1749}{40} \cdot \frac{5}{5} - \frac{1749}{100} \cdot \frac{2}{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.11

Write each expression with a common denominator of $200$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.11.1

Multiply $\frac{1749}{40}$ by $\frac{5}{5}$ .

$\sqrt{\frac{1749 \cdot 5}{40 \cdot 5} - \frac{1749}{100} \cdot \frac{2}{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.11.2

Multiply $40$ by $5$ .

$\sqrt{\frac{1749 \cdot 5}{200} - \frac{1749}{100} \cdot \frac{2}{2}} \frac{20}{\sqrt{17490}}$

Step 8.3.11.3

Multiply $\frac{1749}{100}$ by $\frac{2}{2}$ .

$\sqrt{\frac{1749 \cdot 5}{200} - \frac{1749 \cdot 2}{100 \cdot 2}} \frac{20}{\sqrt{17490}}$

Step 8.3.11.4

Multiply $100$ by $2$ .

$\sqrt{\frac{1749 \cdot 5}{200} - \frac{1749 \cdot 2}{200}} \frac{20}{\sqrt{17490}}$

Step 8.3.12

Combine the numerators over the common denominator.

$\sqrt{\frac{1749 \cdot 5 - 1749 \cdot 2}{200}} \frac{20}{\sqrt{17490}}$

Step 8.3.13

Simplify the numerator.

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Step 8.3.13.1

Multiply $1749$ by $5$ .

$\sqrt{\frac{8745 - 1749 \cdot 2}{200}} \frac{20}{\sqrt{17490}}$

Step 8.3.13.2

Multiply $- 1749$ by $2$ .

$\sqrt{\frac{8745 - 3498}{200}} \frac{20}{\sqrt{17490}}$

Step 8.3.13.3

Subtract $3498$ from $8745$ .

$\sqrt{\frac{5247}{200}} \frac{20}{\sqrt{17490}}$

Step 8.3.14

Rewrite $\sqrt{\frac{5247}{200}}$ as $\frac{\sqrt{5247}}{\sqrt{200}}$ .

$\frac{\sqrt{5247}}{\sqrt{200}} \cdot \frac{20}{\sqrt{17490}}$

Step 8.3.15

Simplify the numerator.

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Step 8.3.15.1

Rewrite $5247$ as $3^{2} \cdot 583$ .

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Step 8.3.15.1.1

Factor $9$ out of $5247$ .

$\frac{\sqrt{9 (583)}}{\sqrt{200}} \cdot \frac{20}{\sqrt{17490}}$

Step 8.3.15.1.2

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

$\frac{\sqrt{3^{2} \cdot 583}}{\sqrt{200}} \cdot \frac{20}{\sqrt{17490}}$

Step 8.3.15.2

Pull terms out from under the radical.

$\frac{3 \sqrt{583}}{\sqrt{200}} \cdot \frac{20}{\sqrt{17490}}$

Step 8.3.16

Simplify the denominator.

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Step 8.3.16.1

Rewrite $200$ as $10^{2} \cdot 2$ .

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Step 8.3.16.1.1

Factor $100$ out of $200$ .

$\frac{3 \sqrt{583}}{\sqrt{100 (2)}} \cdot \frac{20}{\sqrt{17490}}$

Step 8.3.16.1.2

Rewrite $100$ as $10^{2}$ .

$\frac{3 \sqrt{583}}{\sqrt{10^{2} \cdot 2}} \cdot \frac{20}{\sqrt{17490}}$

Step 8.3.16.2

Pull terms out from under the radical.

$\frac{3 \sqrt{583}}{10 \sqrt{2}} \cdot \frac{20}{\sqrt{17490}}$

Step 8.3.17

Simplify terms.

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Step 8.3.17.1

Cancel the common factor of $10$ .

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Step 8.3.17.1.1

Factor $10$ out of $10 \sqrt{2}$ .

$\frac{3 \sqrt{583}}{10 (\sqrt{2})} \cdot \frac{20}{\sqrt{17490}}$

Step 8.3.17.1.2

Factor $10$ out of $20$ .

$\frac{3 \sqrt{583}}{10 (\sqrt{2})} \cdot \frac{10 (2)}{\sqrt{17490}}$

Step 8.3.17.1.3

Cancel the common factor.

$\frac{3 \sqrt{583}}{10 \sqrt{2}} \cdot \frac{10 \cdot 2}{\sqrt{17490}}$

Step 8.3.17.1.4

Rewrite the expression.

$\frac{3 \sqrt{583}}{\sqrt{2}} \cdot \frac{2}{\sqrt{17490}}$

Step 8.3.17.2

Multiply $\frac{3 \sqrt{583}}{\sqrt{2}}$ by $\frac{2}{\sqrt{17490}}$ .

$\frac{3 \sqrt{583} \cdot 2}{\sqrt{2} \sqrt{17490}}$

Step 8.3.17.3

Multiply $2$ by $3$ .

$\frac{6 \sqrt{583}}{\sqrt{2} \sqrt{17490}}$

Step 8.3.17.4

Combine using the product rule for radicals.

$\frac{6 \sqrt{583}}{\sqrt{2 \cdot 17490}}$

Step 8.3.17.5

Multiply $2$ by $17490$ .

$\frac{6 \sqrt{583}}{\sqrt{34980}}$

Step 8.3.18

Combine $\sqrt{583}$ and $\sqrt{34980}$ into a single radical.

$6 \sqrt{\frac{583}{34980}}$

Step 8.3.19

Cancel the common factor of $583$ and $34980$ .

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Step 8.3.19.1

Factor $583$ out of $583$ .

$6 \sqrt{\frac{583 (1)}{34980}}$

Step 8.3.19.2

Cancel the common factors.

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Step 8.3.19.2.1

Factor $583$ out of $34980$ .

$6 \sqrt{\frac{583 \cdot 1}{583 \cdot 60}}$

Step 8.3.19.2.2

Cancel the common factor.

$6 \sqrt{\frac{583 \cdot 1}{583 \cdot 60}}$

Step 8.3.19.2.3

Rewrite the expression.

$6 \sqrt{\frac{1}{60}}$

Step 8.3.20

Rewrite $\sqrt{\frac{1}{60}}$ as $\frac{\sqrt{1}}{\sqrt{60}}$ .

$6 \frac{\sqrt{1}}{\sqrt{60}}$

Step 8.3.21

Any root of $1$ is $1$ .

$6 \frac{1}{\sqrt{60}}$

Step 8.3.22

Simplify the denominator.

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Step 8.3.22.1

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 8.3.22.1.1

Factor $4$ out of $60$ .

$6 \frac{1}{\sqrt{4 (15)}}$

Step 8.3.22.1.2

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

$6 \frac{1}{\sqrt{2^{2} \cdot 15}}$

Step 8.3.22.2

Pull terms out from under the radical.

$6 \frac{1}{2 \sqrt{15}}$

Step 8.3.23

Simplify terms.

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Step 8.3.23.1

Cancel the common factor of $2$ .

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Step 8.3.23.1.1

Factor $2$ out of $6$ .

$2 (3) \frac{1}{2 \sqrt{15}}$

Step 8.3.23.1.2

Factor $2$ out of $2 \sqrt{15}$ .

$2 (3) \frac{1}{2 (\sqrt{15})}$

Step 8.3.23.1.3

Cancel the common factor.

$2 \cdot 3 \frac{1}{2 \sqrt{15}}$

Step 8.3.23.1.4

Rewrite the expression.

$3 \frac{1}{\sqrt{15}}$

Step 8.3.23.2

Combine $3$ and $\frac{1}{\sqrt{15}}$ .

$\frac{3}{\sqrt{15}}$

Step 8.3.24

Multiply $\frac{3}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\frac{3}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}$

Step 8.3.25

Combine and simplify the denominator.

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Step 8.3.25.1

Multiply $\frac{3}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\frac{3 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 8.3.25.2

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{3 \sqrt{15}}{{\sqrt{15}}^{1} \sqrt{15}}$

Step 8.3.25.3

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{3 \sqrt{15}}{{\sqrt{15}}^{1} {\sqrt{15}}^{1}}$

Step 8.3.25.4

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

$\frac{3 \sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Step 8.3.25.5

Add $1$ and $1$ .

$\frac{3 \sqrt{15}}{{\sqrt{15}}^{2}}$

Step 8.3.25.6

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

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Step 8.3.25.6.1

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

$\frac{3 \sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

Step 8.3.25.6.2

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

$\frac{3 \sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

Step 8.3.25.6.3

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

$\frac{3 \sqrt{15}}{15^{\frac{2}{2}}}$

Step 8.3.25.6.4

Cancel the common factor of $2$ .

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Step 8.3.25.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{15}}{15^{\frac{2}{2}}}$

Step 8.3.25.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{15}}{15^{1}}$

Step 8.3.25.6.5

Evaluate the exponent.

$\frac{3 \sqrt{15}}{15}$

Step 8.3.26

Cancel the common factor of $3$ and $15$ .

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Step 8.3.26.1

Factor $3$ out of $3 \sqrt{15}$ .

$\frac{3 (\sqrt{15})}{15}$

Step 8.3.26.2

Cancel the common factors.

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Step 8.3.26.2.1

Factor $3$ out of $15$ .

$\frac{3 \sqrt{15}}{3 \cdot 5}$

Step 8.3.26.2.2

Cancel the common factor.

$\frac{3 \sqrt{15}}{3 \cdot 5}$

Step 8.3.26.2.3

Rewrite the expression.

$\frac{\sqrt{15}}{5}$

Step 9

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

Center: $(\frac{27}{10}, - 4)$

${Vertex}_{1}$ : $(\frac{27}{10}, - 4 + \frac{\sqrt{17490}}{20})$

${Vertex}_{2}$ : $(\frac{27}{10}, - 4 - \frac{\sqrt{17490}}{20})$

${Focus}_{1}$ : $(\frac{27}{10}, - 4 + \frac{3 \sqrt{1166}}{20})$

${Focus}_{2}$ : $(\frac{27}{10}, - 4 - \frac{3 \sqrt{1166}}{20})$

Eccentricity: $\frac{\sqrt{15}}{5}$

Step 10