Find the standard form of the equation of the ellipse satisfying the given conditions.
-Foci: $( 0 , - 2 ) , ( 0,2 ) ; y$-intercepts: $- 5$ and 5
A) $\frac { x ^ { 2 } } { 21 } + \frac { y ^ { 2 } } { 25 } = 1$
B) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 21 } = 1$
C) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 21 } = 1$
D) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 25 } = 1$

Question

Quizplus · Accepted Answer

The standard form of the equation of an ellipse with a vertical major 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 foci is $2c$, where $c^2 = a^2 - b^2$. Given the foci $(0, -2)$ and $(0, 2)$, we find that $2c = 4$, so $c = 2$. The y-intercepts are at $-5$ and $5$, indicating that the major axis length $2a = 10$, so $a = 5$. Using $c^2 = a^2 - b^2$, we get $4 = 25 - b^2$, solving for $b^2$ gives $b^2 = 21$. Therefore, the equation of the ellipse is $\frac{x^2}{21} + \frac{y^2}{25} = 1$.

Find the Standard Form of the Equation of the Ellipse (0,−2),(0,2);y( 0 , - 2 ) , ( 0,2 ) ; y(0,−2),(0,2);y

Find the Standard Form of the Equation of the Ellipse $( 0 , - 2 ) , ( 0,2 ) ; y$