If a circle has the equation x² + y² - 4x + 6y + 9 = 0, what is the center of th

If a circle has the equation x² + y² - 4x + 6y + 9 = 0, what is the center of the circle?

If a circle has the equation x² + y² - 4x + 6y + 9 = 0, what is the center of the circle?

Step 1: Start with the given equation of the circle: x² + y² - 4x + 6y + 9 = 0.
Step 2: Rearrange the equation by moving the constant (9) to the other side: x² + y² - 4x + 6y = -9.
Step 3: Group the x terms and the y terms: (x² - 4x) + (y² + 6y) = -9.
Step 4: Complete the square for the x terms (x² - 4x): Take half of -4 (which is -2), square it (which is 4), and add it inside the parentheses: (x² - 4x + 4) - 4.
Step 5: Complete the square for the y terms (y² + 6y): Take half of 6 (which is 3), square it (which is 9), and add it inside the parentheses: (y² + 6y + 9) - 9.
Step 6: Rewrite the equation with the completed squares: (x - 2)² - 4 + (y + 3)² - 9 = -9.
Step 7: Simplify the equation: (x - 2)² + (y + 3)² = 0.
Step 8: Identify the center of the circle from the standard form (x - h)² + (y - k)² = r², where (h, k) is the center. Here, h = 2 and k = -3.
Step 9: Conclude that the center of the circle is (2, -3).

Circle Equation – Understanding the standard form of a circle's equation and how to convert from general form to standard form.
Completing the Square – The process of rearranging a quadratic equation to identify the center and radius of a circle.