Graph ((x+4)^2)/10+((y-2)^2)/5=1

$\frac{{(x + 4)}^{2}}{10} + \frac{{(y - 2)}^{2}}{5} = 1$

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 $1$ .

$\frac{{(x + 4)}^{2}}{10} + \frac{{(y - 2)}^{2}}{5} = 1$

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.

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

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.

$a = \sqrt{10}$

$b = \sqrt{5}$

$k = 2$

$h = - 4$

Step 4

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(- 4, 2)$

Step 5

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

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

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

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

$\sqrt{{(\sqrt{10})}^{2} - {(\sqrt{5})}^{2}}$

Simplify.

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{{(10^{\frac{1}{2}})}^{2} - {(\sqrt{5})}^{2}}$

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

$\sqrt{10^{\frac{1}{2} \cdot 2} - {(\sqrt{5})}^{2}}$

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

$\sqrt{10^{\frac{2}{2}} - {(\sqrt{5})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{10^{\frac{2}{2}} - {(\sqrt{5})}^{2}}$

Rewrite the expression.

$\sqrt{10^{1} - {(\sqrt{5})}^{2}}$

Evaluate the exponent.

$\sqrt{10 - {(\sqrt{5})}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{10 - {(5^{\frac{1}{2}})}^{2}}$

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

$\sqrt{10 - 5^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{10 - 5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{10 - 5^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{10 - 5^{1}}$

Evaluate the exponent.

$\sqrt{10 - 1 \cdot 5}$

Simplify the expression.

Multiply $- 1$ by $5$ .

$\sqrt{10 - 5}$

Subtract $5$ from $10$ .

$\sqrt{5}$

Step 6

Find the vertices.

The first vertex of an ellipse can be found by adding $a$ to $h$ .

$(h + a, k)$

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

$(- 4 + \sqrt{10}, 2)$

The second vertex of an ellipse can be found by subtracting $a$ from $h$ .

$(h - a, k)$

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

$(- 4 - (\sqrt{10}), 2)$

Simplify.

$(- 4 - \sqrt{10}, 2)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(- 4 + \sqrt{10}, 2)$

${Vertex}_{2}$ : $(- 4 - \sqrt{10}, 2)$

${Vertex}_{1}$ : $(- 4 + \sqrt{10}, 2)$

${Vertex}_{2}$ : $(- 4 - \sqrt{10}, 2)$

Step 7

Find the foci.

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

$(h + c, k)$

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

$(- 4 + \sqrt{5}, 2)$

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

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

$(- 4 - (\sqrt{5}), 2)$

Simplify.

$(- 4 - \sqrt{5}, 2)$

Ellipses have two foci.

${Focus}_{1}$ : $(- 4 + \sqrt{5}, 2)$

${Focus}_{2}$ : $(- 4 - \sqrt{5}, 2)$

${Focus}_{1}$ : $(- 4 + \sqrt{5}, 2)$

${Focus}_{2}$ : $(- 4 - \sqrt{5}, 2)$

Step 8

Find the eccentricity.

Find the eccentricity by using the following formula.

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

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

$\frac{\sqrt{{(\sqrt{10})}^{2} - {(\sqrt{5})}^{2}}}{\sqrt{10}}$

Simplify.

Simplify the numerator.

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{\sqrt{{(10^{\frac{1}{2}})}^{2} - {\sqrt{5}}^{2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10^{\frac{1}{2} \cdot 2} - {\sqrt{5}}^{2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10^{\frac{2}{2}} - {\sqrt{5}}^{2}}}{\sqrt{10}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{10^{\frac{2}{2}} - {\sqrt{5}}^{2}}}{\sqrt{10}}$

Rewrite the expression.

$\frac{\sqrt{10^{1} - {\sqrt{5}}^{2}}}{\sqrt{10}}$

Evaluate the exponent.

$\frac{\sqrt{10 - {\sqrt{5}}^{2}}}{\sqrt{10}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{10 - {(5^{\frac{1}{2}})}^{2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10 - 5^{\frac{1}{2} \cdot 2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10 - 5^{\frac{2}{2}}}}{\sqrt{10}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{10 - 5^{\frac{2}{2}}}}{\sqrt{10}}$

Rewrite the expression.

$\frac{\sqrt{10 - 5^{1}}}{\sqrt{10}}$

Evaluate the exponent.

$\frac{\sqrt{10 - 1 \cdot 5}}{\sqrt{10}}$

Multiply $- 1$ by $5$ .

$\frac{\sqrt{10 - 5}}{\sqrt{10}}$

Subtract $5$ from $10$ .

$\frac{\sqrt{5}}{\sqrt{10}}$

Combine $\sqrt{5}$ and $\sqrt{10}$ into a single radical.

$\sqrt{\frac{5}{10}}$

Cancel the common factor of $5$ and $10$ .

Factor $5$ out of $5$ .

$\sqrt{\frac{5 (1)}{10}}$

Cancel the common factors.

Factor $5$ out of $10$ .

$\sqrt{\frac{5 \cdot 1}{5 \cdot 2}}$

Cancel the common factor.

$\sqrt{\frac{5 \cdot 1}{5 \cdot 2}}$

Rewrite the expression.

$\sqrt{\frac{1}{2}}$

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\sqrt{1}}{\sqrt{2}}$

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

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

$\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\frac{\sqrt{2}}{2}$

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(- 4, 2)$

${Vertex}_{1}$ : $(- 4 + \sqrt{10}, 2)$

${Vertex}_{2}$ : $(- 4 - \sqrt{10}, 2)$

${Focus}_{1}$ : $(- 4 + \sqrt{5}, 2)$

${Focus}_{2}$ : $(- 4 - \sqrt{5}, 2)$

Eccentricity: $\frac{\sqrt{2}}{2}$

Step 10

Graph ((x+4)^2)/10+((y-2)^2)/5=1

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