Find an equation for the ellipse with center at the origin, major axis of length $2 \sqrt { 7 }$ , minor axis of length $2 \sqrt { 3 }$
, and whose foci lie on the $y$
-axis
A) $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 7 } = 1$

B) $\frac { x ^ { 2 } } { 12 } + \frac { y ^ { 2 } } { 14 } = 1$

C) $\frac { x ^ { 2 } } { 14 } + \frac { y ^ { 2 } } { 12 } = 1$

D) $\frac { x ^ { 2 } } { 7 } + \frac { y ^ { 2 } } { 3 } = 1$

E) none of these

Question

Find an equation for the ellipse with center at the origin, major axis of length $2 \sqrt { 7 }$ , minor axis of length $2 \sqrt { 3 }$
, and whose foci lie on the $y$
-axis
A) $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 7 } = 1$

B) $\frac { x ^ { 2 } } { 12 } + \frac { y ^ { 2 } } { 14 } = 1$

C) $\frac { x ^ { 2 } } { 14 } + \frac { y ^ { 2 } } { 12 } = 1$

D) $\frac { x ^ { 2 } } { 7 } + \frac { y ^ { 2 } } { 3 } = 1$

E) none of these

Quizplus · Accepted Answer

The equation of an ellipse with the center at the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $2a$ is the length of the major axis and $2b$ is the length of the minor axis. Since the foci lie on the $y$-axis, the major axis is along the $y$-axis, making $2a = 2\sqrt{7}$ and $2b = 2\sqrt{3}$. Therefore, $a^2 = 7$ and $b^2 = 3$, leading to the equation $\frac{x^2}{3} + \frac{y^2}{7} = 1$.

Find an Equation for the Ellipse with Center at the Origin