» » » x$$^2$$ + 20x + y$$^2$$ + 16y = -20The equation above defines a circle in the xy...

# 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)

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).