Question

The equation x2+y2−4x+2y=b describes a circle. Complete the square to find the center of the circle. The center of the circle is ( , ) The radius of the circle is 7 units. What is the value of b in the equation? b=

Answer 1

Look to the coefficients on the middle terms (-4x and 2y)

divide each coefficient by 2 and square to determine what to add.

(-4/2)^2 = 4 needs to be added to complete the square for x.

(2/2)^2 = 1 needs to be added to complete the square for y.

add to both sides of the equal sign to keep the equation balanced.

x^2 - 4x + 4 + y^2 + 2y + 1 = b + 4 + 1
factor
(x - 2)^2 + (y + 1)^2 = b + 5

The general equation for a circle is:
(x - h)^2 + (y - k)^2 = r^2
with center (h,k) and radius r

compare this to the equation. you see that the center is: (2, -1)

we can write
r^2 = b + 5
given that r = 7
49 = b + 5
49 - 5 = b
44 = b