Answer
$\frac{x^2}{36} + \frac{y^2}{11} = 1$
Work Step by Step
RECALL:
The standard equation of an ellipse with whose center is at (0, 0) is:
(i) $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ (horizontal major axis)
(ii) $\frac{x^2}{b^2} + \frac{y^2}{a^2}=1$ (vertical major axis)
where
$a \gt b$
2a = length of major axis
2b = length of minor axis
c = distance of each focus from the center
$a^2=b^2+c^2$
The foci are (-5, 0) and (5, 0), which are equidistant from the origin (0, 0). This means that the ellipse has its center at (0, 0).
The foci, when connect together, form a horizontal segment. This means that the major axis is also horizontal.
The length of major axis is 12 $\longrightarrow 2a=12 \longrightarrow a=6$
Each focus is 5 units away from the center so $c=5$.
Solve for $b$ using the formula $a^2=b^2=c^2$ to have:
$\\a^2 = b^2+c^2
\\6^2=b^2+5^2
\\36=b^2+25
\\36-25=b^2
\\11=b^2
\\\sqrt{11}=b$
Therefore, the equation of the ellipse is:
$\\\frac{x^2}{a^2} + \frac{y^2}{b^2}=1
\\\frac{x^2}{6^2} + \frac{y^2}{(\sqrt{11})^2}=1
\\\frac{x^2}{36} + \frac{y^2}{11} = 1$
