Question

If \( (-3, 2) \) lies on the circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \), which is concentric with the circle \( x^2 + y^2 + 6x + 8y - 5 = 0 \), then \( c \) is equal to:

• A. 11
• B. -11
• C. 24
• D. 100

Hint:

When two circles are concentric, they have the same center. Use this property to relate the constants \( g \) and \( f \) of both circles and solve for \( c \).

Answer 1

The Correct Option is B

Step 1: Understanding the given equations.
We are given two circles. The general equation of a circle is: \[ x^2 + y^2 + 2gx + 2fy + c = 0, \] where \( (g, f) \) is the center of the circle and \( c \) is a constant. The second circle is: \[ x^2 + y^2 + 6x + 8y - 5 = 0. \] This is a standard circle equation where we can identify the center and radius.
Step 2: Finding the center of the second circle.
Rewrite the second equation by completing the square: \[ x^2 + 6x + y^2 + 8y = 5. \] Complete the square for \( x \) and \( y \): \[ (x^2 + 6x) = (x + 3)^2 - 9 \quad \text{and} \quad (y^2 + 8y) = (y + 4)^2 - 16. \] Substitute these into the equation: \[ (x + 3)^2 - 9 + (y + 4)^2 - 16 = 5. \] Simplify: \[ (x + 3)^2 + (y + 4)^2 = 30. \] So, the center of this circle is \( (-3, -4) \) and the radius is \( \sqrt{30} \).
Step 3: The center of the first circle.
The first circle is concentric with the second circle, meaning it has the same center, which is \( (-3, -4) \). So, the center of the first circle is \( (-3, -4) \).
Step 4: Using the point \( (-3, 2) \) on the first circle.
The point \( (-3, 2) \) lies on the first circle, so substitute \( x = -3 \) and \( y = 2 \) into the equation of the first circle: \[ (-3)^2 + (2)^2 + 2g(-3) + 2f(2) + c = 0. \] Simplify: \[ 9 + 4 - 6g + 4f + c = 0, \] \[ 13 - 6g + 4f + c = 0. \] This equation represents the relationship between \( g, f, \) and \( c \).
Step 5: Comparing the coefficients of the concentric circles.
Since the first and second circles are concentric, the values of \( g \) and \( f \) in the first circle must be equal to those in the second circle. From the second circle, we know that: \[ g = -3 \quad \text{and} \quad f = -4. \] Substitute these values into the equation: \[ 13 - 6(-3) + 4(-4) + c = 0, \] \[ 13 + 18 - 16 + c = 0, \] \[ 15 + c = 0, \] \[ c = -11. \]
Step 6: Conclusion.
Thus, the value of \( c \) is \( -11 \), and the correct answer is (b).