A circle is represented by the equation x2+y2−16x−6y=−24 . Complete the square to discover the center and radius of the circle.
To complete the square for x, we first rearrange the equation:
x^2 - 16x + y^2 - 6y = -24
Now, we will complete the square for x by adding and subtracting (16/2)^2 = 64:
x^2 - 16x + 64 + y^2 - 6y = -24 + 64
(x - 8)^2 + y^2 - 6y = 40
Next, we will complete the square for y by adding and subtracting (6/2)^2 = 9:
(x - 8)^2 + y^2 - 6y + 9 = 40 + 9
(x - 8)^2 + (y - 3)^2 = 49
Now, the equation is in the standard form of a circle:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) is the center of the circle and r is the radius. Therefore, the center of the circle is at (h, k) = (8, 3) and the radius is sqrt(49) = 7.
So, the center of the circle is (8, 3) and the radius is 7.