Question

The center and radius for the circle \( x^2 + y^2 + 6x - 4y + 4 = 0 \) respectively are:

• A. (2, 3) and 3
• B. (3, 2) and 8
• C. (2, -3) and 3
• D. (-3, 2) and 3

Hint:

To complete the square for circles, add and subtract the required constants to make perfect squares of the terms with \( x \) and \( y \).

Answer 1

The Correct Option is C

Step 1: Rewrite the equation of the circle in standard form.
The given equation is \( x^2 + y^2 + 6x - 4y + 4 = 0 \). We can complete the square for both \( x \) and \( y \). - For \( x^2 + 6x \), we complete the square by adding and subtracting \( 9 \): \[ x^2 + 6x = (x + 3)^2 - 9 \] - For \( y^2 - 4y \), we complete the square by adding and subtracting \( 4 \): \[ y^2 - 4y = (y - 2)^2 - 4 \] So, the equation becomes: \[ (x + 3)^2 - 9 + (y - 2)^2 - 4 + 4 = 0 \] Simplifying: \[ (x + 3)^2 + (y - 2)^2 = 9 \]

Step 2: Find the center and radius.
The equation is now in the form \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius. Thus, the center is \( (-3, 2) \) and the radius is \( \sqrt{9} = 3 \). Therefore, the correct answer is 3. (2, -3) and 3.