Find the standard form of the equation of the ellipse satisfying the given conditions.
-Foci: $( - 3,0 ) , ( 3,0 ) ; x$-intercepts: $- 4$ and 4
A) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$
B) $\frac { x ^ { 2 } } { 7 } + \frac { y ^ { 2 } } { 16 } = 1$
C) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 7 } = 1$
D) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$

Question

Quizplus · Accepted Answer

The standard form of an ellipse with a horizontal major axis 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 between the foci is $2c$, where $c^2 = a^2 - b^2$. Given foci at $(-3,0)$ and $(3,0)$, we have $2c = 6$, so $c = 3$. The x-intercepts at $-4$ and $4$ indicate that the major axis length is $8$, so $a = 4$. Thus, $c^2 = 9$ and $a^2 = 16$, leading to $b^2 = a^2 - c^2 = 16 - 9 = 7$. Therefore, the equation is $\frac{x^2}{16} + \frac{y^2}{7} = 1$.

Find the Standard Form of the Equation of the Ellipse (−3,0),(3,0);x( - 3,0 ) , ( 3,0 ) ; x(−3,0),(3,0);x

Find the Standard Form of the Equation of the Ellipse $( - 3,0 ) , ( 3,0 ) ; x$