Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.
Vertices: $( \pm 7,0 )$ ; Foci: $( \pm 5,0 )$
A) $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 24 } = 1$
B) $\frac { x ^ { 2 } } { 7 } + \frac { y ^ { 2 } } { 5 } = 1$
C) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 24 } = 1$
D) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 25 } = 1$
E) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 24 } = 1$

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 
Vertices: $( \pm 7,0 )$ ; Foci: $( \pm 5,0 )$ 
A) $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 24 } = 1$
B) $\frac { x ^ { 2 } } { 7 } + \frac { y ^ { 2 } } { 5 } = 1$
C) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 24 } = 1$
D) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 25 } = 1$
E) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 24 } = 1$

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 $2a$ is the length of the major axis and $2b$ is the length of the minor axis. The vertices given ($\pm 7,0$) indicate that the major axis is along the x-axis and $a = 7$, so $a^2 = 49$. The foci being at $(\pm 5,0)$ means $c = 5$, where $c^2 = a^2 - b^2$. Solving for $b^2$ gives $b^2 = a^2 - c^2 = 49 - 25 = 24$. Therefore, the correct equation is $\frac{x^2}{49} + \frac{y^2}{24} = 1$.

Find the Standard Form of the Equation of the Ellipse (±7,0)( \pm 7,0 )(±7,0)

Find the Standard Form of the Equation of the Ellipse $( \pm 7,0 )$