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x\(^2\) + 20x + y\(^2\) + 16y = -20The equation above defines a circle in the xy...


Question

x\(^2\) + 20x + y\(^2\) + 16y = -20

The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle? 

Options

A)
(−20, −16) 
B)
(−10, −8) 
C)
(10, 8) 
D)
(20, 16)

The correct answer is B.

Explanation:

Choice B is correct. The standard equation of a circle in the xy-plane is of the form (x − h)\(^2\) + (y − k)\(^2\) = r\(^2\), where (h, k) are the coordinates of the center of the circle and r is the radius. To convert the given equation to the standard form, complete the squares. The first two terms need a 100 to complete the square, and the second two terms need a 64. Adding 100 and 64 to both sides of the given equation yields (x\(^2\) + 20x + 100) + (y\(^2\) + 16y + 64) = −20 + 100 + 64, which is equivalent to (x + 10)\(^2\) + (y + 8)\(^2\) = 144. Therefore, the coordinates of the center of the circle are (−10, −8).

Choice A is incorrect and is likely the result of not properly dividing when attempting to complete the square. Choice C is incorrect and is likely the result of making a sign error when evaluating the coordinates of the center. Choice D is incorrect and is likely the result of not properly dividing when attempting to complete the square and making a sign error when evaluating the coordinates of the center.


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