Final answer:
The correct answer is option c. The equation given represents an ellipse, not a circle. After completing the square, the center of the ellipse is determined to be at (4, -10). However, the equation does not provide a single radius, as an ellipse has two axes, not a radius.
Step-by-step explanation:
The equation x²/8+y²/20-8x+20y=-80 does not, in fact, describe a circle, but rather an ellipse due to the differing coefficients for x² and y² terms. To find the center and radius of a circle, the equation should be in the standard form of (x-h)²+(y-k)²=r². However, we can complete the square for both x and y terms to find the center of an ellipse.
For the x terms: x² - 8x can be written as (x - 4)² - 16 after completing the square. For the y terms: y² + 20y can be written as (y + 10)² - 100 after completing the square. The equation then becomes:
(x-4)²/8 + (y+10)²/20 - 16 - 100 = -80
By bringing the constants to the right side and simplifying, we have:
(x-4)²/8 + (y+10)²/20 = 36
Now, we can see the center of the ellipse is at (4, -10) and the correct answer for the center of the ellipse is option c) (4,-10). However, the question incorrectly asks for a radius, which does not apply to an ellipse. The values given after the coordinates are not radii; rather, the numbers represent the squares of the semi-major and semi-minor axes for the ellipse equation. In any case, this mistake means that none of the answer choices completely and correctly describes the ellipse’s characteristics.