Precalculus Examples

$\frac{{(x + 3)}^{2}}{7} + \frac{{(y - 4)}^{2}}{16} = 1$

Step 1

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 $1$ .

$\frac{{(x + 3)}^{2}}{7} + \frac{{(y - 4)}^{2}}{16} = 1$

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.

$a = 4$

$b = \sqrt{7}$

$k = 4$

$h = - 3$

Step 4

Find the eccentricity by using the following formula.

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

Step 5

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

$\frac{\sqrt{{(4)}^{2} - {(\sqrt{7})}^{2}}}{4}$

Step 6

Simplify the numerator.

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Step 6.1

Raise $4$ to the power of $2$ .

$\frac{\sqrt{16 - {\sqrt{7}}^{2}}}{4}$

Step 6.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{16 - {(7^{\frac{1}{2}})}^{2}}}{4}$

Step 6.2.2

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

$\frac{\sqrt{16 - 7^{\frac{1}{2} \cdot 2}}}{4}$

Step 6.2.3

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

$\frac{\sqrt{16 - 7^{\frac{2}{2}}}}{4}$

Step 6.2.4

Cancel the common factor of $2$ .

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Step 6.2.4.1

Cancel the common factor.

$\frac{\sqrt{16 - 7^{\frac{2}{2}}}}{4}$

Step 6.2.4.2

Rewrite the expression.

$\frac{\sqrt{16 - 7^{1}}}{4}$

Step 6.2.5

Evaluate the exponent.

$\frac{\sqrt{16 - 1 \cdot 7}}{4}$

Step 6.3

Multiply $- 1$ by $7$ .

$\frac{\sqrt{16 - 7}}{4}$

Step 6.4

Subtract $7$ from $16$ .

$\frac{\sqrt{9}}{4}$

Step 6.5

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

$\frac{\sqrt{3^{2}}}{4}$

Step 6.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{4}$

Step 7