Find the Properties ((x+2)^2)/100+((y+7)^2)/36=1

$\frac{{(x + 2)}^{2}}{100} + \frac{{(y + 7)}^{2}}{36} = 1$

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 $1$ .

$\frac{{(x + 2)}^{2}}{100} + \frac{{(y + 7)}^{2}}{36} = 1$

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.

$a = 10$

$b = 6$

$k = - 7$

$h = - 2$

Step 4

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

$(- 2, - 7)$

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.

$\sqrt{{(10)}^{2} - {(6)}^{2}}$

Simplify.

Raise $10$ to the power of $2$ .

$\sqrt{100 - {(6)}^{2}}$

Raise $6$ to the power of $2$ .

$\sqrt{100 - 1 \cdot 36}$

Multiply $- 1$ by $36$ .

$\sqrt{100 - 36}$

Subtract $36$ from $100$ .

$\sqrt{64}$

Rewrite $64$ as $8^{2}$ .

$\sqrt{8^{2}}$

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

$8$

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.

$(- 2 + 10, - 7)$

Simplify.

$(8, - 7)$

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.

$(- 2 - (10), - 7)$

Simplify.

$(- 12, - 7)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(8, - 7)$

${Vertex}_{2}$ : $(- 12, - 7)$

${Vertex}_{1}$ : $(8, - 7)$

${Vertex}_{2}$ : $(- 12, - 7)$

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.

$(- 2 + 8, - 7)$

Simplify.

$(6, - 7)$

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.

$(- 2 - (8), - 7)$

Simplify.

$(- 10, - 7)$

Ellipses have two foci.

${Focus}_{1}$ : $(6, - 7)$

${Focus}_{2}$ : $(- 10, - 7)$

${Focus}_{1}$ : $(6, - 7)$

${Focus}_{2}$ : $(- 10, - 7)$

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.

$\frac{\sqrt{{(10)}^{2} - {(6)}^{2}}}{10}$

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$\frac{\sqrt{100 - 6^{2}}}{10}$

Raise $6$ to the power of $2$ .

$\frac{\sqrt{100 - 1 \cdot 36}}{10}$

Multiply $- 1$ by $36$ .

$\frac{\sqrt{100 - 36}}{10}$

Subtract $36$ from $100$ .

$\frac{\sqrt{64}}{10}$

Rewrite $64$ as $8^{2}$ .

$\frac{\sqrt{8^{2}}}{10}$

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

$\frac{8}{10}$

Cancel the common factor of $8$ and $10$ .

Factor $2$ out of $8$ .

$\frac{2 (4)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{2 \cdot 4}{2 \cdot 5}$

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 5}$

Rewrite the expression.

$\frac{4}{5}$

Step 9

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

Center: $(- 2, - 7)$

${Vertex}_{1}$ : $(8, - 7)$

${Vertex}_{2}$ : $(- 12, - 7)$

${Focus}_{1}$ : $(6, - 7)$

${Focus}_{2}$ : $(- 10, - 7)$

Eccentricity: $\frac{4}{5}$

Step 10

Find the Properties ((x+2)^2)/100+((y+7)^2)/36=1

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