Calculus Examples

$r^{2} = a^{2} \cos (2 θ)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $a^{2} \cos (2 θ)$ .

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Step 1.1.1

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

$r^{2} = a^{2} (2 \cos^{2} (θ) - 1)$

Step 1.1.2

Simplify by multiplying through.

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Step 1.1.2.1

Apply the distributive property.

$r^{2} = a^{2} (2 \cos^{2} (θ)) + a^{2} \cdot - 1$

Step 1.1.2.2

Reorder.

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Step 1.1.2.2.1

Rewrite using the commutative property of multiplication.

$r^{2} = 2 a^{2} \cos^{2} (θ) + a^{2} \cdot - 1$

Step 1.1.2.2.2

Move $- 1$ to the left of $a^{2}$ .

$r^{2} = 2 a^{2} \cos^{2} (θ) - 1 \cdot a^{2}$

Step 1.1.3

Rewrite $- 1 a^{2}$ as $- a^{2}$ .

$r^{2} = 2 a^{2} \cos^{2} (θ) - a^{2}$

Step 2

Since $\cos (θ) = \frac{x}{r}$ , replace $\cos (θ)$ with $\frac{x}{r}$ .

$r^{2} = 2 a^{2} {(\frac{x}{r})}^{2} - a^{2}$

Step 3

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} = 2 a^{2} {(\frac{x}{\sqrt{x^{2} + y^{2}}})}^{2} - a^{2}$

Step 4

Write in standard form.

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Step 4.1

Move all the expressions to the left side of the equation.

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Step 4.1.1

Subtract $2 a^{2} {(\frac{x}{\sqrt{x^{2} + y^{2}}})}^{2}$ from both sides of the equation.

$x^{2} + y^{2} - 2 a^{2} {(\frac{x}{\sqrt{x^{2} + y^{2}}})}^{2} = - a^{2}$

Step 4.1.2

Add $a^{2}$ to both sides of the equation.

$x^{2} + y^{2} - 2 a^{2} {(\frac{x}{\sqrt{x^{2} + y^{2}}})}^{2} + a^{2} = 0$

Step 4.2

Simplify $x^{2} + y^{2} - 2 a^{2} {(\frac{x}{\sqrt{x^{2} + y^{2}}})}^{2} + a^{2}$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $\frac{x}{\sqrt{x^{2} + y^{2}}}$ by $\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$ .

$x^{2} + y^{2} - 2 a^{2} {(\frac{x}{\sqrt{x^{2} + y^{2}}} \cdot \frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}})}^{2} + a^{2} = 0$

Step 4.2.1.2

Combine and simplify the denominator.

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Step 4.2.1.2.1

Multiply $\frac{x}{\sqrt{x^{2} + y^{2}}}$ by $\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$ .

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}} \sqrt{x^{2} + y^{2}}})}^{2} + a^{2} = 0$

Step 4.2.1.2.2

Raise $\sqrt{x^{2} + y^{2}}$ to the power of $1$ .

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{1} \sqrt{x^{2} + y^{2}}})}^{2} + a^{2} = 0$

Step 4.2.1.2.3

Raise $\sqrt{x^{2} + y^{2}}$ to the power of $1$ .

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{1} {\sqrt{x^{2} + y^{2}}}^{1}})}^{2} + a^{2} = 0$

Step 4.2.1.2.4

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

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{1 + 1}})}^{2} + a^{2} = 0$

Step 4.2.1.2.5

Add $1$ and $1$ .

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{2}})}^{2} + a^{2} = 0$

Step 4.2.1.2.6

Rewrite ${\sqrt{x^{2} + y^{2}}}^{2}$ as $x^{2} + y^{2}$ .

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Step 4.2.1.2.6.1

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

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{{({(x^{2} + y^{2})}^{\frac{1}{2}})}^{2}})}^{2} + a^{2} = 0$

Step 4.2.1.2.6.2

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

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{\frac{1}{2} \cdot 2}})}^{2} + a^{2} = 0$

Step 4.2.1.2.6.3

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

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{\frac{2}{2}}})}^{2} + a^{2} = 0$

Step 4.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.1.2.6.4.1

Cancel the common factor.

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{\frac{2}{2}}})}^{2} + a^{2} = 0$

Step 4.2.1.2.6.4.2

Rewrite the expression.

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{1}})}^{2} + a^{2} = 0$

Step 4.2.1.2.6.5

Simplify.

$x^{2} + y^{2} - 2 a^{2} {(\frac{x \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}})}^{2} + a^{2} = 0$

Step 4.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.1.3.1

Apply the product rule to $\frac{x \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$ .

$x^{2} + y^{2} - 2 a^{2} \frac{{(x \sqrt{x^{2} + y^{2}})}^{2}}{{(x^{2} + y^{2})}^{2}} + a^{2} = 0$

Step 4.2.1.3.2

Apply the product rule to $x \sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} - 2 a^{2} \frac{x^{2} {\sqrt{x^{2} + y^{2}}}^{2}}{{(x^{2} + y^{2})}^{2}} + a^{2} = 0$

Step 4.2.1.4

Rewrite ${\sqrt{x^{2} + y^{2}}}^{2}$ as $x^{2} + y^{2}$ .

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Step 4.2.1.4.1

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

$x^{2} + y^{2} - 2 a^{2} \frac{x^{2} {({(x^{2} + y^{2})}^{\frac{1}{2}})}^{2}}{{(x^{2} + y^{2})}^{2}} + a^{2} = 0$

Step 4.2.1.4.2

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

$x^{2} + y^{2} - 2 a^{2} \frac{x^{2} {(x^{2} + y^{2})}^{\frac{1}{2} \cdot 2}}{{(x^{2} + y^{2})}^{2}} + a^{2} = 0$

Step 4.2.1.4.3

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

$x^{2} + y^{2} - 2 a^{2} \frac{x^{2} {(x^{2} + y^{2})}^{\frac{2}{2}}}{{(x^{2} + y^{2})}^{2}} + a^{2} = 0$

Step 4.2.1.4.4

Cancel the common factor of $2$ .

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Step 4.2.1.4.4.1

Cancel the common factor.

$x^{2} + y^{2} - 2 a^{2} \frac{x^{2} {(x^{2} + y^{2})}^{\frac{2}{2}}}{{(x^{2} + y^{2})}^{2}} + a^{2} = 0$

Step 4.2.1.4.4.2

Rewrite the expression.

$x^{2} + y^{2} - 2 a^{2} \frac{x^{2} {(x^{2} + y^{2})}^{1}}{{(x^{2} + y^{2})}^{2}} + a^{2} = 0$

Step 4.2.1.4.5

Simplify.

$x^{2} + y^{2} - 2 a^{2} \frac{x^{2} (x^{2} + y^{2})}{{(x^{2} + y^{2})}^{2}} + a^{2} = 0$

Step 4.2.1.5

Cancel the common factor of $x^{2} + y^{2}$ and ${(x^{2} + y^{2})}^{2}$ .

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Step 4.2.1.5.1

Factor $x^{2} + y^{2}$ out of $x^{2} (x^{2} + y^{2})$ .

$x^{2} + y^{2} - 2 a^{2} \frac{(x^{2} + y^{2}) x^{2}}{{(x^{2} + y^{2})}^{2}} + a^{2} = 0$

Step 4.2.1.5.2

Cancel the common factors.

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Step 4.2.1.5.2.1

Factor $x^{2} + y^{2}$ out of ${(x^{2} + y^{2})}^{2}$ .

$x^{2} + y^{2} - 2 a^{2} \frac{(x^{2} + y^{2}) x^{2}}{(x^{2} + y^{2}) (x^{2} + y^{2})} + a^{2} = 0$

Step 4.2.1.5.2.2

Cancel the common factor.

$x^{2} + y^{2} - 2 a^{2} \frac{(x^{2} + y^{2}) x^{2}}{(x^{2} + y^{2}) (x^{2} + y^{2})} + a^{2} = 0$

Step 4.2.1.5.2.3

Rewrite the expression.

$x^{2} + y^{2} - 2 a^{2} \frac{x^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.1.6

Multiply $- 2 a^{2} \frac{x^{2}}{x^{2} + y^{2}}$ .

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Step 4.2.1.6.1

Combine $- 2$ and $\frac{x^{2}}{x^{2} + y^{2}}$ .

$x^{2} + y^{2} + a^{2} \frac{- 2 x^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.1.6.2

Combine $a^{2}$ and $\frac{- 2 x^{2}}{x^{2} + y^{2}}$ .

$x^{2} + y^{2} + \frac{a^{2} (- 2 x^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.1.7

Remove unnecessary parentheses.

$x^{2} + y^{2} + \frac{a^{2} \cdot - 2 x^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.1.8

Move $- 2$ to the left of $a^{2}$ .

$x^{2} + y^{2} + \frac{- 2 \cdot a^{2} x^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.1.9

Move the negative in front of the fraction.

$x^{2} + y^{2} - \frac{2 a^{2} x^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + y^{2}}{x^{2} + y^{2}}$ .

$y^{2} + \frac{x^{2} (x^{2} + y^{2})}{x^{2} + y^{2}} - \frac{2 a^{2} x^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.3

Combine the numerators over the common denominator.

$y^{2} + \frac{x^{2} (x^{2} + y^{2}) - 2 a^{2} x^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.4

Factor $x^{2}$ out of $x^{2} (x^{2} + y^{2}) - 2 a^{2} x^{2}$ .

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Step 4.2.4.1

Factor $x^{2}$ out of $- 2 a^{2} x^{2}$ .

$y^{2} + \frac{x^{2} (x^{2} + y^{2}) + x^{2} (- 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.4.2

Factor $x^{2}$ out of $x^{2} (x^{2} + y^{2}) + x^{2} (- 2 a^{2})$ .

$y^{2} + \frac{x^{2} (x^{2} + y^{2} - 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.5

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + y^{2}}{x^{2} + y^{2}}$ .

$\frac{y^{2} (x^{2} + y^{2})}{x^{2} + y^{2}} + \frac{x^{2} (x^{2} + y^{2} - 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.6

Combine the numerators over the common denominator.

$\frac{y^{2} (x^{2} + y^{2}) + x^{2} (x^{2} + y^{2} - 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.7

Simplify the numerator.

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Step 4.2.7.1

Apply the distributive property.

$\frac{y^{2} x^{2} + y^{2} y^{2} + x^{2} (x^{2} + y^{2} - 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.7.2

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 4.2.7.2.1

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

$\frac{y^{2} x^{2} + y^{2 + 2} + x^{2} (x^{2} + y^{2} - 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.7.2.2

Add $2$ and $2$ .

$\frac{y^{2} x^{2} + y^{4} + x^{2} (x^{2} + y^{2} - 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.7.3

Apply the distributive property.

$\frac{y^{2} x^{2} + y^{4} + x^{2} x^{2} + x^{2} y^{2} + x^{2} (- 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.7.4

Simplify.

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Step 4.2.7.4.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 4.2.7.4.1.1

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

$\frac{y^{2} x^{2} + y^{4} + x^{2 + 2} + x^{2} y^{2} + x^{2} (- 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.7.4.1.2

Add $2$ and $2$ .

$\frac{y^{2} x^{2} + y^{4} + x^{4} + x^{2} y^{2} + x^{2} (- 2 a^{2})}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.7.4.2

Rewrite using the commutative property of multiplication.

$\frac{y^{2} x^{2} + y^{4} + x^{4} + x^{2} y^{2} - 2 x^{2} a^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.7.5

Add $y^{2} x^{2}$ and $x^{2} y^{2}$ .

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Step 4.2.7.5.1

Reorder $y^{2}$ and $x^{2}$ .

$\frac{y^{4} + x^{4} + x^{2} y^{2} + x^{2} y^{2} - 2 x^{2} a^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.7.5.2

Add $x^{2} y^{2}$ and $x^{2} y^{2}$ .

$\frac{y^{4} + x^{4} + 2 x^{2} y^{2} - 2 x^{2} a^{2}}{x^{2} + y^{2}} + a^{2} = 0$

Step 4.2.8

To write $a^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + y^{2}}{x^{2} + y^{2}}$ .

$\frac{y^{4} + x^{4} + 2 x^{2} y^{2} - 2 x^{2} a^{2}}{x^{2} + y^{2}} + \frac{a^{2} (x^{2} + y^{2})}{x^{2} + y^{2}} = 0$

Step 4.2.9

Combine the numerators over the common denominator.

$\frac{y^{4} + x^{4} + 2 x^{2} y^{2} - 2 x^{2} a^{2} + a^{2} (x^{2} + y^{2})}{x^{2} + y^{2}} = 0$

Step 4.2.10

Simplify the numerator.

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Step 4.2.10.1

Apply the distributive property.

$\frac{y^{4} + x^{4} + 2 x^{2} y^{2} - 2 x^{2} a^{2} + a^{2} x^{2} + a^{2} y^{2}}{x^{2} + y^{2}} = 0$

Step 4.2.10.2

Add $- 2 x^{2} a^{2}$ and $a^{2} x^{2}$ .

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Step 4.2.10.2.1

Move $x^{2}$ .

$\frac{y^{4} + x^{4} + 2 x^{2} y^{2} - 2 a^{2} x^{2} + a^{2} x^{2} + a^{2} y^{2}}{x^{2} + y^{2}} = 0$

Step 4.2.10.2.2

Add $- 2 a^{2} x^{2}$ and $a^{2} x^{2}$ .

$\frac{y^{4} + x^{4} + 2 x^{2} y^{2} - a^{2} x^{2} + a^{2} y^{2}}{x^{2} + y^{2}} = 0$

Step 5