Precalculus Examples

$35 x^{2} + y^{2} = 35$

Step 1

Divide each term by $35$ to make the right side equal to one.

$\frac{35 x^{2}}{35} + \frac{y^{2}}{35} = \frac{35}{35}$

Step 2

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 $1$ .

$x^{2} + \frac{y^{2}}{35} = 1$

Step 3

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 4

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.

$a = \sqrt{35}$

$b = 1$

$k = 0$

$h = 0$

Step 5

Find the eccentricity by using the following formula.

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

Step 6

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

$\frac{\sqrt{{(\sqrt{35})}^{2} - {(1)}^{2}}}{\sqrt{35}}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 7.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\frac{\sqrt{{(35^{\frac{1}{2}})}^{2} - 1^{2}}}{\sqrt{35}}$

Step 7.1.1.2

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

$\frac{\sqrt{35^{\frac{1}{2} \cdot 2} - 1^{2}}}{\sqrt{35}}$

Step 7.1.1.3

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

$\frac{\sqrt{35^{\frac{2}{2}} - 1^{2}}}{\sqrt{35}}$

Step 7.1.1.4

Cancel the common factor of $2$ .

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Step 7.1.1.4.1

Cancel the common factor.

$\frac{\sqrt{35^{\frac{2}{2}} - 1^{2}}}{\sqrt{35}}$

Step 7.1.1.4.2

Rewrite the expression.

$\frac{\sqrt{35^{1} - 1^{2}}}{\sqrt{35}}$

Step 7.1.1.5

Evaluate the exponent.

$\frac{\sqrt{35 - 1^{2}}}{\sqrt{35}}$

Step 7.1.2

One to any power is one.

$\frac{\sqrt{35 - 1 \cdot 1}}{\sqrt{35}}$

Step 7.1.3

Multiply $- 1$ by $1$ .

$\frac{\sqrt{35 - 1}}{\sqrt{35}}$

Step 7.1.4

Subtract $1$ from $35$ .

$\frac{\sqrt{34}}{\sqrt{35}}$

Step 7.2

Multiply $\frac{\sqrt{34}}{\sqrt{35}}$ by $\frac{\sqrt{35}}{\sqrt{35}}$ .

$\frac{\sqrt{34}}{\sqrt{35}} \cdot \frac{\sqrt{35}}{\sqrt{35}}$

Step 7.3

Combine and simplify the denominator.

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Step 7.3.1

Multiply $\frac{\sqrt{34}}{\sqrt{35}}$ by $\frac{\sqrt{35}}{\sqrt{35}}$ .

$\frac{\sqrt{34} \sqrt{35}}{\sqrt{35} \sqrt{35}}$

Step 7.3.2

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{\sqrt{34} \sqrt{35}}{{\sqrt{35}}^{1} \sqrt{35}}$

Step 7.3.3

Raise $\sqrt{35}$ to the power of $1$ .

$\frac{\sqrt{34} \sqrt{35}}{{\sqrt{35}}^{1} {\sqrt{35}}^{1}}$

Step 7.3.4

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

$\frac{\sqrt{34} \sqrt{35}}{{\sqrt{35}}^{1 + 1}}$

Step 7.3.5

Add $1$ and $1$ .

$\frac{\sqrt{34} \sqrt{35}}{{\sqrt{35}}^{2}}$

Step 7.3.6

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 7.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\frac{\sqrt{34} \sqrt{35}}{{(35^{\frac{1}{2}})}^{2}}$

Step 7.3.6.2

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

$\frac{\sqrt{34} \sqrt{35}}{35^{\frac{1}{2} \cdot 2}}$

Step 7.3.6.3

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

$\frac{\sqrt{34} \sqrt{35}}{35^{\frac{2}{2}}}$

Step 7.3.6.4

Cancel the common factor of $2$ .

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Step 7.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{34} \sqrt{35}}{35^{\frac{2}{2}}}$

Step 7.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{34} \sqrt{35}}{35^{1}}$

Step 7.3.6.5

Evaluate the exponent.

$\frac{\sqrt{34} \sqrt{35}}{35}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Combine using the product rule for radicals.

$\frac{\sqrt{34 \cdot 35}}{35}$

Step 7.4.2

Multiply $34$ by $35$ .

$\frac{\sqrt{1190}}{35}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{1190}}{35}$

Decimal Form:

$0.98561076 \dots$

Step 9