Trigonometry Examples

$25 x^{2} + 4 y^{2} + 100 x = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Complete the square for $25 x^{2} + 100 x$ .

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Step 1.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 25$

$b = 100$

$c = 0$

Step 1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{100}{2 \cdot 25}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of $100$ and $2$ .

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Step 1.1.3.2.1.1

Factor $2$ out of $100$ .

$d = \frac{2 \cdot 50}{2 \cdot 25}$

Step 1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.3.2.1.2.1

Factor $2$ out of $2 \cdot 25$ .

$d = \frac{2 \cdot 50}{2 (25)}$

Step 1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 50}{2 \cdot 25}$

Step 1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{50}{25}$

Step 1.1.3.2.2

Cancel the common factor of $50$ and $25$ .

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Step 1.1.3.2.2.1

Factor $25$ out of $50$ .

$d = \frac{25 \cdot 2}{25}$

Step 1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.3.2.2.2.1

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 2}{25 (1)}$

Step 1.1.3.2.2.2.2

Cancel the common factor.

$d = \frac{25 \cdot 2}{25 \cdot 1}$

Step 1.1.3.2.2.2.3

Rewrite the expression.

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

Step 1.1.3.2.2.2.4

Divide $2$ by $1$ .

$d = 2$

Step 1.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{100^{2}}{4 \cdot 25}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Raise $100$ to the power of $2$ .

$e = 0 - \frac{10000}{4 \cdot 25}$

Step 1.1.4.2.1.2

Multiply $4$ by $25$ .

$e = 0 - \frac{10000}{100}$

Step 1.1.4.2.1.3

Divide $10000$ by $100$ .

$e = 0 - 1 \cdot 100$

Step 1.1.4.2.1.4

Multiply $- 1$ by $100$ .

$e = 0 - 100$

Step 1.1.4.2.2

Subtract $100$ from $0$ .

$e = - 100$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(x + 2)}^{2} - 100$ .

$25 {(x + 2)}^{2} - 100$

Step 1.2

Substitute $25 {(x + 2)}^{2} - 100$ for $25 x^{2} + 100 x$ in the equation $25 x^{2} + 4 y^{2} + 100 x = 0$ .

$25 {(x + 2)}^{2} - 100 + 4 y^{2} = 0$

Step 1.3

Move $- 100$ to the right side of the equation by adding $100$ to both sides.

$25 {(x + 2)}^{2} + 4 y^{2} = 0 + 100$

Step 1.4

Add $0$ and $100$ .

$25 {(x + 2)}^{2} + 4 y^{2} = 100$

Step 1.5

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

$\frac{25 {(x + 2)}^{2}}{100} + \frac{4 y^{2}}{100} = \frac{100}{100}$

Step 1.6

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 $1$ .

$\frac{{(x + 2)}^{2}}{4} + \frac{y^{2}}{25} = 1$

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.

$a = 5$

$b = 2$

$k = 0$

$h = - 2$

Step 4

Find $c$ , the distance from the center to a focus.

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Step 4.1

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Step 4.2

Substitute the values of $a$ and $b$ in the formula.

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

Step 4.3

Simplify.

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Step 4.3.1

Raise $5$ to the power of $2$ .

$\sqrt{25 - {(2)}^{2}}$

Step 4.3.2

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

$\sqrt{25 - 1 \cdot 4}$

Step 4.3.3

Multiply $- 1$ by $4$ .

$\sqrt{25 - 4}$

Step 4.3.4

Subtract $4$ from $25$ .

$\sqrt{21}$

Step 5

Find the foci.

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Step 5.1

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Step 5.2

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

$(- 2, 0 + \sqrt{21})$

Step 5.3

Simplify.

$(- 2, \sqrt{21})$

Step 5.4

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

$(h, k - c)$

Step 5.5

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

$(- 2, 0 - (\sqrt{21}))$

Step 5.6

Simplify.

$(- 2, - \sqrt{21})$

Step 5.7

Ellipses have two foci.

${Focus}_{1}$ : $(- 2, \sqrt{21})$

${Focus}_{2}$ : $(- 2, - \sqrt{21})$

${Focus}_{1}$ : $(- 2, \sqrt{21})$

${Focus}_{2}$ : $(- 2, - \sqrt{21})$

Step 6