Find the standard form of the equation of the ellipse with the given characteristics.
Foci: ; major axis of length 12.

Question

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci:  ; major axis of length 12.
​

Quizplus · Accepted Answer

The given foci are not in standard form, so we need to identify the center of the ellipse first. The center is halfway between the two foci, which would be (0,0) in standard form.

The length of the major axis is given as 12, so the distance from the center to each end of the major axis is 6.

The distance between the foci (c) is given as: 
c = sqrt(a^2-b^2)

where a is the distance from the center to each end of the major axis (6 in this case), and b is the distance from the center to each end of the minor axis.

We don't know b yet, but we can use the fact that the length of the major axis is 12 to find it: 
2a = 12 
a = 6 
sqrt(a^2-b^2) = 11

Now we can solve for b: 
121 = 36 - b^2 
b^2 = -85

This is not possible, since b^2 cannot be negative. Therefore, there is no real solution and the given characteristics do not correspond to a valid ellipse.

None of the answer choices are correct, but the closest one is C, which is the only choice that includes the correct foci.

Find the Standard Form of the Equation of the Ellipse