Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.
Foci: ; major axis of length 12

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​
Foci:  ; major axis of length 12
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Quizplus · Accepted Answer

The standard form of the equation of an ellipse centered at the origin is:
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
where $a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis.
The distance between the foci is $2\sqrt{a^2-b^2}=12$.
Since the center is at the origin and $c=\sqrt{a^2-b^2}$ for an ellipse, we know that $c=\frac{12}{2}=6$. 
We also know that one focus is at $(6,0)$, and since the other focus is the same distance away, it must be at $(-6,0)$.
Therefore, $c^2=36=a^2-b^2$.
Since the major axis has length 12, $2a=12$ or $a=6$.
Substituting $a$ and $c^2$ into the equation above yields:
$\frac{x^2}{36}+\frac{y^2}{36-36}=1$
Simplifying gives:
$\frac{x^2}{36}=1$
Or, in standard form:
$x^2=36$.
The correct answer is D because $x^2=36$ is only one option.

Find the Standard Form of the Equation of the Ellipse