Solve the problem.
-The ellipse $\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 144 } = 1$ is shifted horizontally and vertically to obtain the ellipse $\frac { ( x - 3 ) ^ { 2 } } { 400 } + \frac { ( y + 3 ) ^ { 2 } } { 144 } = 1$ Find the foci of the new ellipse.
A) (-3, -9), (-3, 15)
B) (-9, -3), (15, -3)
C) (-13, -3), (19, -3)
D) (-3, -13), (-3, 19)

Question

Quizplus · Accepted Answer

To find the foci of the shifted ellipse, we need to find the center of the shifted ellipse first. From the equations of the two ellipses, we can see that the center of the shifted ellipse is at (3,-3), which means that the ellipse has been shifted 3 units to the right and 3 units downwards.

Now, we can find the foci of the shifted ellipse using the formula:

c = sqrt(a^2 - b^2),

where a and b are the semi-major and semi-minor axes of the ellipse, respectively. For the shifted ellipse, we have a = 20 and b = 12. Thus,

c = sqrt(20^2 - 12^2) = sqrt(256) = 16.

Therefore, the foci of the shifted ellipse are located at a distance of 16 units from the center in the horizontal direction.

The center of the ellipse is (3,-3), so the foci are located at (-13,-3) and (19,-3).

Therefore, the answer is choice C, (-13,-3) and (19,-3).

Solve the Problem x2400+y2144=1\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 144 } = 1400x2​+144y2​=1

Solve the Problem $\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 144 } = 1$