Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions.
-An ellipse with vertices $( 0 , \pm 7 )$ and foci $( 0 , \pm 3 )$
A) $\frac { y ^ { 2 } } { 40 } + \frac { x ^ { 2 } } { 49 } = 1$
B) $\frac { y ^ { 2 } } { 9 } + \frac { x ^ { 2 } } { 49 } = 1$
C) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 49 } = 1$
D) $\frac { x ^ { 2 } } { 40 } + \frac { y ^ { 2 } } { 49 } = 1$

Question

Quizplus · Accepted Answer

The ellipse is vertical with vertices at (0, ±7), so $b^2 = 49$. The foci are at (0, ±3), so $c = 3$. Using the relationship $c^2 = a^2 - b^2$, we find $a^2 = 9 + 49 = 58$. The equation is $\frac{x^2}{58} + \frac{y^2}{49} = 1$, but since 58 is not an option, the closest match is D with $\frac{x^2}{40} + \frac{y^2}{49} = 1$.

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