Algebra Examples

\frac{x^{2}}{73} - \frac{y^{2}}{19} = 1

$\frac{x^{2}}{73} - \frac{y^{2}}{19} = 1$

Step 1

Simplify each term in the equation in order to set the right side equal to

1

$1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be

1

$1$ .

\frac{x^{2}}{73} - \frac{y^{2}}{19} = 1

$\frac{x^{2}}{73} - \frac{y^{2}}{19} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 3

Match the values in this hyperbola to those of the standard form. The variable

h

$h$ represents the x-offset from the origin,

k

$k$ represents the y-offset from origin,

a

$a$ .

a = \sqrt{73}

$a = \sqrt{73}$

b = \sqrt{19}

$b = \sqrt{19}$

k = 0

$k = 0$

h = 0

$h = 0$

Step 4

Find

c

$c$ , 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 hyperbola by using the following formula.

\sqrt{a^{2} + b^{2}}

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

Step 4.2

Substitute the values of

a

$a$ and

b

$b$ in the formula.

\sqrt{{(\sqrt{73})}^{2} + {(\sqrt{19})}^{2}}

$\sqrt{{(\sqrt{73})}^{2} + {(\sqrt{19})}^{2}}$

Step 4.3

Simplify.

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Step 4.3.1

Rewrite

{\sqrt{73}}^{2}

${\sqrt{73}}^{2}$ as

73

$73$ .

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Step 4.3.1.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{73}

$\sqrt{73}$ as

73^{\frac{1}{2}}

$73^{\frac{1}{2}}$ .

\sqrt{{(73^{\frac{1}{2}})}^{2} + {(\sqrt{19})}^{2}}

$\sqrt{{(73^{\frac{1}{2}})}^{2} + {(\sqrt{19})}^{2}}$

Step 4.3.1.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\sqrt{73^{\frac{1}{2} \cdot 2} + {(\sqrt{19})}^{2}}

$\sqrt{73^{\frac{1}{2} \cdot 2} + {(\sqrt{19})}^{2}}$

Step 4.3.1.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{73^{\frac{2}{2}} + {(\sqrt{19})}^{2}}

$\sqrt{73^{\frac{2}{2}} + {(\sqrt{19})}^{2}}$

Step 4.3.1.4

Cancel the common factor of

2

$2$ .

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Step 4.3.1.4.1

Cancel the common factor.

\sqrt{73^{\frac{2}{2}} + {(\sqrt{19})}^{2}}

$\sqrt{73^{\frac{2}{2}} + {(\sqrt{19})}^{2}}$

Step 4.3.1.4.2

Rewrite the expression.

\sqrt{73^{1} + {(\sqrt{19})}^{2}}

$\sqrt{73^{1} + {(\sqrt{19})}^{2}}$

\sqrt{73^{1} + {(\sqrt{19})}^{2}}

$\sqrt{73^{1} + {(\sqrt{19})}^{2}}$

Step 4.3.1.5

Evaluate the exponent.

\sqrt{73 + {(\sqrt{19})}^{2}}

$\sqrt{73 + {(\sqrt{19})}^{2}}$

\sqrt{73 + {(\sqrt{19})}^{2}}

$\sqrt{73 + {(\sqrt{19})}^{2}}$

Step 4.3.2

Rewrite

{\sqrt{19}}^{2}

${\sqrt{19}}^{2}$ as

19

$19$ .

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Step 4.3.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{19}

$\sqrt{19}$ as

19^{\frac{1}{2}}

$19^{\frac{1}{2}}$ .

\sqrt{73 + {(19^{\frac{1}{2}})}^{2}}

$\sqrt{73 + {(19^{\frac{1}{2}})}^{2}}$

Step 4.3.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\sqrt{73 + 19^{\frac{1}{2} \cdot 2}}

$\sqrt{73 + 19^{\frac{1}{2} \cdot 2}}$

Step 4.3.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{73 + 19^{\frac{2}{2}}}

$\sqrt{73 + 19^{\frac{2}{2}}}$

Step 4.3.2.4

Cancel the common factor of

2

$2$ .

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Step 4.3.2.4.1

Cancel the common factor.

\sqrt{73 + 19^{\frac{2}{2}}}

$\sqrt{73 + 19^{\frac{2}{2}}}$

Step 4.3.2.4.2

Rewrite the expression.

\sqrt{73 + 19^{1}}

$\sqrt{73 + 19^{1}}$

\sqrt{73 + 19^{1}}

$\sqrt{73 + 19^{1}}$

Step 4.3.2.5

Evaluate the exponent.

\sqrt{73 + 19}

$\sqrt{73 + 19}$

\sqrt{73 + 19}

$\sqrt{73 + 19}$

Step 4.3.3

Add

73

$73$ and

19

$19$ .

\sqrt{92}

$\sqrt{92}$

Step 4.3.4

Rewrite

92

$92$ as

2^{2} \cdot 23

$2^{2} \cdot 23$ .

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Step 4.3.4.1

Factor

4

$4$ out of

92

$92$ .

\sqrt{4 (23)}

$\sqrt{4 (23)}$

Step 4.3.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

\sqrt{2^{2} \cdot 23}

$\sqrt{2^{2} \cdot 23}$

\sqrt{2^{2} \cdot 23}

$\sqrt{2^{2} \cdot 23}$

Step 4.3.5

Pull terms out from under the radical.

2 \sqrt{23}

$2 \sqrt{23}$

2 \sqrt{23}

$2 \sqrt{23}$

2 \sqrt{23}

$2 \sqrt{23}$

Step 5

Find the foci.

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Step 5.1

The first focus of a hyperbola can be found by adding

c

$c$ to

h

$h$ .

(h + c, k)

$(h + c, k)$

Step 5.2

Substitute the known values of

h

$h$ ,

c

$c$ , and

k

$k$ into the formula and simplify.

(2 \sqrt{23}, 0)

$(2 \sqrt{23}, 0)$

Step 5.3

The second focus of a hyperbola can be found by subtracting

c

$c$ from

h

$h$ .

(h - c, k)

$(h - c, k)$

Step 5.4

Substitute the known values of

h

$h$ ,

c

$c$ , and

k

$k$ into the formula and simplify.

(- 2 \sqrt{23}, 0)

$(- 2 \sqrt{23}, 0)$

Step 5.5

The foci of a hyperbola follow the form of

(h \pm \sqrt{a^{2} + b^{2}}, k)

$(h \pm \sqrt{a^{2} + b^{2}}, k)$ . Hyperbolas have two foci.

(2 \sqrt{23}, 0), (- 2 \sqrt{23}, 0)

$(2 \sqrt{23}, 0), (- 2 \sqrt{23}, 0)$

(2 \sqrt{23}, 0), (- 2 \sqrt{23}, 0)

$(2 \sqrt{23}, 0), (- 2 \sqrt{23}, 0)$

Step 6