Provide an appropriate response. -Which of the following inequalities satisfies the following description: the region inside an ellipse centered at the origin, with $x$-intercepts at $( - 8,0 )$ and $( 8,0 )$, and $y$-intercepts at $( 0 , - 4 )$ and $( 0,4 )$ ? A) $\frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 4 } 1$ C) $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 16 }

Question

Provide an appropriate response.
-Which of the following inequalities satisfies the following description: the region inside an ellipse centered at the origin, with $x$-intercepts at $( - 8,0 )$ and $( 8,0 )$, and $y$-intercepts at $( 0 , - 4 )$ and $( 0,4 )$ ?
A) $\frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 4 } < 1$
B) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } > 1$
C) $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 16 } < 1$
D) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } < 1$

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 $2a$ and $2b$ are the lengths of the major and minor axes, respectively. Given the $x$-intercepts are $(-8, 0)$ and $(8, 0)$, the length of the major axis is $16$, so $a = 8$. Given the $y$-intercepts are $(0, -4)$ and $(0, 4)$, the length of the minor axis is $8$, so $b = 4$. Therefore, the equation becomes $\frac{x^2}{64} + \frac{y^2}{16} < 1$ for the region inside the ellipse.

Provide an Appropriate Response xxx -Intercepts At (−8,0)( - 8,0 )(−8,0) And (8,0)( 8,0 )(8,0)

Provide an Appropriate Response $x$ -Intercepts At $( - 8,0 )$ And $( 8,0 )$