Final answer:
To put the equation in standard form, group the x terms and the y terms separately, and complete the square for each variable. The equation x^2 + y^2 + 10x - 6y + 33 = 0 can be written in standard form as (x + 5)^2 + (y - 3)^2 = 1.
Step-by-step explanation:
The given equation is x^2 + y^2 + 10x - 6y + 33 = 0. To put it in standard form, we need to group the x terms and the y terms separately, and complete the square for each variable.
Moving the constant term to the right side, we have x^2 + 10x + y^2 - 6y = -33. Next, to complete the square for x, we take half of the coefficient of x (which is 10) and square it, giving us (10/2)^2 = 25. Add 25 to both sides of the equation: x^2 + 10x + 25 + y^2 - 6y = -33 + 25.
Similarly, to complete the square for y, we take half of the coefficient of y (which is -6) and square it, giving us (-6/2)^2 = 9. Add 9 to both sides of the equation: x^2 + 10x + 25 + y^2 - 6y + 9 = -33 + 25 + 9.
Simplifying, we have (x + 5)^2 + (y - 3)^2 = 1.
Therefore, the equation in standard form is (x + 5)^2 + (y - 3)^2 = 1.