Rewrite the following equation in the center-radius form of the equation of a circle.
x2 - 2x + y2 + 2y - 2 = 0
just complete the squares as needed, and balance out what you had to change:
x^2-2x+1 + y^2+2y+1 - 2 - (1+1) = 0
(x-1)^2 + (y+1)^2 = 4
To rewrite the equation in the center-radius form of the equation of a circle, we need to complete the square for both the x and y terms.
For the x terms:
1. Rewrite the equation by factoring out the x² and x terms:
(x² - 2x) + (y² + 2y) - 2 = 0
2. Complete the square for the x terms by adding (2/2)² = 1 to both sides:
(x² - 2x + 1) + (y² + 2y) - 2 + 1 = 1
3. Simplify the equation:
(x - 1)² + (y² + 2y) - 1 = 1
For the y terms:
4. Complete the square for the y terms by adding (2/2)² = 1 to both sides:
(x - 1)² + (y² + 2y + 1) - 1 + 1 = 1 + 1
5. Simplify the equation:
(x - 1)² + (y + 1)² = 2
Now, the equation is in the center-radius form of the equation of a circle:
(x - 1)² + (y + 1)² = 2
To rewrite the equation in the center-radius form of the equation of a circle, we need to complete the square for both the x-terms and the y-terms.
Starting with the x-terms:
1. Take half of the coefficient of 'x' (which is -2): -2/2 = -1
2. Square this value: (-1)^2 = 1
3. Add this result to both sides of the equation:
x^2 - 2x + 1 + y^2 + 2y - 2 = 1
(x^2 - 2x + 1) + (y^2 + 2y - 2) = 1
Now, let's complete the square for the y-terms:
1. Take half of the coefficient of 'y' (which is 2): 2/2 = 1
2. Square this value: (1)^2 = 1
3. Add this result to both sides of the equation:
(x^2 - 2x + 1) + (y^2 + 2y + 1) - 2 = 1 + 1
(x^2 - 2x + 1) + (y^2 + 2y + 1) = 4
Now, we can rewrite this equation in the center-radius form by grouping the x-terms and y-terms separately:
1. Group the x-terms and complete the square:
(x - 1)^2 + (y^2 + 2y + 1) = 4
2. Group the y-terms and complete the square:
(x - 1)^2 + (y + 1)^2 = 4
Finally, we can see that the equation is now in the center-radius form:
(x - 1)^2 + (y + 1)^2 = 4
The center of the circle is at the point (1, -1) and the radius of the circle is 2.