Find the Foci ((x+4)^2)/100+((y+5)^2)/25=1

$\frac{{(x + 4)}^{2}}{100} + \frac{{(y + 5)}^{2}}{25} = 1$

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 $1$ .

$\frac{{(x + 4)}^{2}}{100} + \frac{{(y + 5)}^{2}}{25} = 1$

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.

$a = 10$

$b = 5$

$k = - 5$

$h = - 4$

Step 4

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.

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

Simplify.

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

$\sqrt{100 - {(5)}^{2}}$

Raise $5$ to the power of $2$ .

$\sqrt{100 - 1 \cdot 25}$

Multiply $- 1$ by $25$ .

$\sqrt{100 - 25}$

Subtract $25$ from $100$ .

$\sqrt{75}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$\sqrt{25 (3)}$

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$5 \sqrt{3}$

Step 5

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.

$(- 4 + 5 \sqrt{3}, - 5)$

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.

$(- 4 - (5 \sqrt{3}), - 5)$

Simplify.

$(- 4 - 5 \sqrt{3}, - 5)$

Ellipses have two foci.

${Focus}_{1}$ : $(- 4 + 5 \sqrt{3}, - 5)$

${Focus}_{2}$ : $(- 4 - 5 \sqrt{3}, - 5)$

${Focus}_{1}$ : $(- 4 + 5 \sqrt{3}, - 5)$

${Focus}_{2}$ : $(- 4 - 5 \sqrt{3}, - 5)$

Step 6

Find the Foci ((x+4)^2)/100+((y+5)^2)/25=1

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