Find an equation for the ellipse described.
-Center at $( 0,0 )$; focus at $( 0,5 )$; vertex at $( 0 , - 7 )$
A) $\frac { x ^ { 2 } } { 24 } + \frac { y ^ { 2 } } { 49 } = 1$
B) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 24 } = 1$
C) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 49 } = 1$
D) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 24 } = 1$

Question

Quizplus · Accepted Answer

The equation 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 distance from the center to a focus is $c$, and these values satisfy the equation $c^2 = a^2 - b^2$. Given that the focus is at $(0, 5)$, we have $c = 5$, and given that the vertex is at $(0, -7)$, we have $a = 7$. Since the ellipse is oriented along the y-axis (the longer axis is vertical), $a^2 = 49$ and $b^2$ can be found using $c^2 = a^2 - b^2$, which gives $b^2 = a^2 - c^2 = 49 - 25 = 24$. Therefore, the correct equation is $\frac{x^2}{24} + \frac{y^2}{49} = 1$.

Find an Equation for the Ellipse Described (0,0)( 0,0 )(0,0) ; Focus At

Find an Equation for the Ellipse Described $( 0,0 )$ ; Focus At