Question

If the radius of a circle \( x^2 + y^2 - 4x + 6y - k = 0 \) is 5, then \( k = \)

• A. \( -12 \)
• B. \( -25 \)
• C. \( 25 \)
• D. \( 12 \)

Hint:

When completing the square for the equation of a circle, remember to adjust the constant to maintain the correct equation.

Answer 1

The Correct Option is D

Step 1: Rewrite the equation of the circle.
The equation of the circle is \( x^2 + y^2 - 4x + 6y - k = 0 \). Completing the square for \( x \) and \( y \), we get: \[ (x - 2)^2 + (y + 3)^2 = 25 \] Thus, the radius of the circle is 5, and we find \( k = 12 \).
Step 2: Conclusion.
The correct answer is (D) \( 12 \).