The equation x2 + y2 − 2x + 2y − 1 = 0 is the general form of the equation of a circle. What is the standard form of the equation?
x^2 + y^2 − 2x + 2y − 1 = 0 , notice how we indicate exponents
complete the square
x^2 + y^2 − 2x + 2y = 1
x^2 - 2x + 1 + y^2 + 2y + 1 = 1+1+1
(x-1)^2 + (y+1)^2 = 3
To find the standard form of the equation of a circle, we need to rewrite the given equation in a specific format. The standard form of the equation of a circle is:
(x - h)2 + (y - k)2 = r2
To convert the given equation to the standard form, we need to complete the square for both the x and y terms.
Given equation: x2 + y2 − 2x + 2y − 1 = 0
Step 1: Group the x's and y's together.
x2 - 2x + y2 + 2y = 1
Step 2: Complete the square for the x terms by adding the square of half the coefficient of x, which is (2/2)^2 = 1.
x2 - 2x + 1 + y2 + 2y = 1 + 1
Step 3: Complete the square for the y terms by adding the square of half the coefficient of y, which is (2/2)^2 = 1.
x2 - 2x + 1 + y2 + 2y + 1 = 1 + 1 + 1
Step 4: Simplify the equation.
(x - 1)2 + (y + 1)2 = 3
The equation (x - 1)2 + (y + 1)2 = 3 is now in the standard form.
Therefore, the standard form of the equation is (x - 1)2 + (y + 1)2 = 3.