Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions.
-An ellipse with vertices $( 0 , \pm 9 )$ and foci at $( 0 , \pm 2 \sqrt { 3 } )$.
A) $\frac { x ^ { 2 } } { 69 } + \frac { y ^ { 2 } } { 81 } = 1$
B) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 69 } = 1$
C) $\frac { x ^ { 2 } } { 69 } + \frac { y ^ { 2 } } { 93 } = 1$
D) $\frac { x ^ { 2 } } { 12 } + \frac { y ^ { 2 } } { 81 } = 1$

Question

Quizplus · Accepted Answer

The standard form of an ellipse centered at the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $2a$ and $2b$ are the lengths of the major and minor axes, respectively. Given vertices at $(0, \pm 9)$, $b = 9$, so $b^2 = 81$. The foci are at $(0, \pm 2\sqrt{3})$, so $c = 2\sqrt{3}$, and $c^2 = 12$. Since $c^2 = b^2 - a^2$ for ellipses with vertical major axes, solving for $a^2$ gives $a^2 = b^2 - c^2 = 81 - 12 = 69$. Therefore, the equation is $\frac{x^2}{69} + \frac{y^2}{81} = 1$.

Find the Standard-Form Equation of the Ellipse Centered at the Origin