Question

Using the equation x2+y2−8x+2y=8 , calculate the center and radius of the circle by completing the square.

Answer 1

To complete the square for the given equation, we need to rewrite it in the form of:

(x-h)2 + (y-k)2 = r2

Where (h,k) is the center of the circle and r is the radius.

Given equation: x2 + y2 - 8x + 2y = 8

Rearrange terms to group the x and y terms together:

(x2 - 8x) + (y2 + 2y) = 8

To complete the square for x terms:
1. Take half of the coefficient of x (-8) and square it: (-8/2)2 = 16
2. Add and subtract this value inside the parentheses:

(x2 - 8x + 16 - 16) + (y2 + 2y) = 8
(x-4)2 - 16 + (y2 + 2y) = 8

To complete the square for y terms:
1. Take half of the coefficient of y (2) and square it: (2/2)2 = 1
2. Add and subtract this value inside the parentheses:

(x-4)2 - 16 + (y2 + 2y + 1 - 1) = 8
(x-4)2 + (y+1)2 - 16 - 1 = 8
(x-4)2 + (y+1)2 = 25

Now, the equation is in the form of (x-h)2 + (y-k)2 = r2, where the center is at (h,k) = (4,-1) and the radius r = √25 = 5.

Therefore, the center of the circle is (4,-1) and the radius of the circle is 5.