Find the standard form of the equation of the ellipse satisfying the given conditions.
-Foci: $( 0 , - 4 ) , ( 0,4 )$; vertices: $( 0 , - 7 ) , ( 0,7 )$
A) $\frac { x ^ { 2 } } { 33 } + \frac { y ^ { 2 } } { 49 } = 1$
B) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$
C) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 33 } = 1$
D) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 49 } = 1$

Question

Quizplus · Accepted Answer

The standard form of the equation of an ellipse with its major axis along the y-axis is $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$, where $2a$ is the length of the major axis and $2b$ is the length of the minor axis. The distance between the center and a focus is $c$, and $c^2 = a^2 - b^2$. Given that the vertices are at $(0, -7)$ and $(0, 7)$, the length of the major axis is $2a = 14$, so $a = 7$ and $a^2 = 49$. The foci are at $(0, -4)$ and $(0, 4)$, so $c = 4$ and $c^2 = 16$. Using $c^2 = a^2 - b^2$, we find $b^2 = a^2 - c^2 = 49 - 16 = 33$. Thus, the equation of the ellipse is $\frac{x^2}{33} + \frac{y^2}{49} = 1$.

Find the Standard Form of the Equation of the Ellipse (0,−4),(0,4)( 0 , - 4 ) , ( 0,4 )(0,−4),(0,4)

Find the Standard Form of the Equation of the Ellipse $( 0 , - 4 ) , ( 0,4 )$