Write an equation for the ellipse.
-foci at $( \pm 2 \sqrt { 15 } , 0 ) ; x$-intercepts $\pm 8$
A) $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 8 } = 1$
B) $\frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 2 } = 1$
C) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 4 } = 1$
D) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 64 } = 1$

Question

Quizplus · Accepted Answer

The equation of an ellipse in standard form 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 center and a focus is $c$, and these values are related by $c^2 = a^2 - b^2$. Given the foci at $(\pm 2\sqrt{15}, 0)$, we have $c = 2\sqrt{15}$. The x-intercepts at $\pm 8$ indicate that the major axis is along the x-axis, and $2a = 16$, so $a = 8$. Using $c^2 = a^2 - b^2$, we find $b^2 = a^2 - c^2 = 64 - 4 \cdot 15 = 64 - 60 = 4$. Thus, the equation is $\frac{x^2}{64} + \frac{y^2}{4} = 1$.

Write an Equation for the Ellipse (±215,0);x( \pm 2 \sqrt { 15 } , 0 ) ; x(±215​,0);x

Write an Equation for the Ellipse $( \pm 2 \sqrt { 15 } , 0 ) ; x$