Find the standard-form equation for an ellipse which satisfies the given conditions.
-An ellipse centered at the origin having vertex at $( 0 , - 9 )$ and eccentricity equal to $\frac { 1 } { 3 }$
A) $\frac { y ^ { 2 } } { 81 } + \frac { x ^ { 2 } } { 72 } = 1$
B) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 72 } = 1$
C) $\frac { y ^ { 2 } } { 81 } - \frac { x ^ { 2 } } { 72 } = 1$
D) $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 72 } = 1$

Question

Quizplus · Accepted Answer

The given vertex indicates the major axis is along the y-axis, making $a=9$ (since the vertex distance from the center is $a$). The eccentricity $e=\frac{1}{3}$ and $a=9$ give $c=\frac{a}{e}=3$. Using $c^2=a^2-b^2$, we find $b^2=72$, leading to the equation $\frac{y^2}{81}+\frac{x^2}{72}=1$.

Find the Standard-Form Equation for an Ellipse Which Satisfies the Given