Find the Foci ((x-43)^2)/225+((y-68)^2)/169=1

$\frac{{(x - 43)}^{2}}{225} + \frac{{(y - 68)}^{2}}{169} = 1$

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 $1$ .

$\frac{{(x - 43)}^{2}}{225} + \frac{{(y - 68)}^{2}}{169} = 1$

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.

$a = 15$

$b = 13$

$k = 68$

$h = 43$

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.

$\sqrt{{(15)}^{2} - {(13)}^{2}}$

Simplify.

Raise $15$ to the power of $2$ .

$\sqrt{225 - {(13)}^{2}}$

Raise $13$ to the power of $2$ .

$\sqrt{225 - 1 \cdot 169}$

Multiply $- 1$ by $169$ .

$\sqrt{225 - 169}$

Subtract $169$ from $225$ .

$\sqrt{56}$

Rewrite $56$ as $2^{2} \cdot 14$ .

Factor $4$ out of $56$ .

$\sqrt{4 (14)}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 14}$

Pull terms out from under the radical.

$2 \sqrt{14}$

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.

$(43 + 2 \sqrt{14}, 68)$

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.

$(43 - (2 \sqrt{14}), 68)$

Simplify.

$(43 - 2 \sqrt{14}, 68)$

Ellipses have two foci.

${Focus}_{1}$ : $(43 + 2 \sqrt{14}, 68)$

${Focus}_{2}$ : $(43 - 2 \sqrt{14}, 68)$

${Focus}_{1}$ : $(43 + 2 \sqrt{14}, 68)$

${Focus}_{2}$ : $(43 - 2 \sqrt{14}, 68)$

Step 6

Find the Foci ((x-43)^2)/225+((y-68)^2)/169=1

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