Question
Using the equation x2+y2−8x+2y=8 , calculate the center and radius of the circle by completing the square.
Answers
GPT 3.5
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.
(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.