Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.
Foci: $( \pm 5,0 )$ ;major axis of length 12

A) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 11 } = 1$
B) $\frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 5 } = 1$
C) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 25 } = 1$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 11 } = 1$
E) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 11 } = 1$

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​
Foci: $( \pm 5,0 )$ ;major axis of length 12
​
A) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 11 } = 1$
B) $\frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 5 } = 1$
C) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 25 } = 1$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 11 } = 1$
E) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 11 } = 1$

Quizplus · Accepted Answer

Since the foci and center are at the origin, we know that the center is (0,0). We also know that the major axis has length 12, so the distance from the center to the edge of the ellipse along the major axis is 6. Since the foci lie on the major axis, we know that the distance from the center to each focus is $\frac{1}{2}$ of the major axis length, which is 6/2=3.

Therefore, we can use the standard form equation for an ellipse centered at the origin:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

where a is the length of the semi-major axis and b is the length of the semi-minor axis. Since the distance from the center to the edge of the ellipse along the major axis is 6, we have a=6. Also, since the distance from the center to each focus is 3, which is $\sqrt{a^2-b^2}$, we can solve for b:

$3=\sqrt{a^2-b^2}$
$3=\sqrt{6^2-b^2}$
$3=\sqrt{36-b^2}$
$9=36-b^2$
$b^2=27$

Therefore, the equation of the ellipse is:

$\frac{x^2}{36}+\frac{y^2}{27}=1$

which is in standard form. Choice D is the only answer that matches this equation.

Find the Standard Form of the Equation of the Ellipse (±5,0)( \pm 5,0 )(±5,0)

Find the Standard Form of the Equation of the Ellipse $( \pm 5,0 )$