If the circumference of the circle `x^2 + y^2 + 8x + 8y - b = 0` is bisected by the circle `x^2 + y^2 - 2x + 4y + a = 0` then `a+b=` (A) 50 (B) 56 (C) `-56` (D) `-34`

Options

A. 50
B. 56
C. `-56`
D. `-34`

Explanation

To solve the problem, we need to analyze the two circles given in the equations and find the values of \( a \) and \( b \) such that the circumference of one circle is bisected by the other.

### Step 1: Identify the equations of the circles
The equations of the circles are:
1. Circle \( S_1: x^2 + y^2 + 8x + 8y - b = 0 \)
2. Circle \( S_2: x^2 + y^2 - 2x + 4y + a = 0 \)

### Step 2: Rewrite the equations in standard form
To rewrite these equations in standard form, we complete the square for both circles.

**For Circle \( S_1 \)**:
- Rearranging gives:
\[ x^2 + 8x + y^2 + 8y = b \]
- Completing the square:
\[ (x + 4)^2 - 16 + (y + 4)^2 - 16 = b \]
- Thus, we have:
\[ (x + 4)^2 + (y + 4)^2 = b + 32 \]
- Center of \( S_1 \) is \( (-4, -4) \) and radius is \( \sqrt{b + 32} \).

**For Circle \( S_2 \)**:
- Rearranging gives:
\[ x^2 - 2x + y^2 + 4y = -a \]
- Completing the square:
\[ (x - 1)^2 - 1 + (y + 2)^2 - 4 = -a \]
- Thus, we have:
\[ (x - 1)^2 + (y + 2)^2 = -a + 5 \]
- Center of \( S_2 \) is \( (1, -2) \) and radius is \( \sqrt{-a + 5} \).

### Step 3: Find the equation of the common chord
Since the circumference of circle \( S_1 \) is bisected by circle \( S_2 \), the equation of the common chord can be expressed as:
\[ S_1 - S_2 = 0 \]
Substituting the equations:
\[ (x^2 + y^2 + 8x + 8y - b) - (x^2 + y^2 - 2x + 4y + a) = 0 \]
This simplifies to:
\[ 10x + 4y - (a + b) = 0 \]

### Step 4: Substitute the center of \( S_1 \) into the common chord equation
The center of circle \( S_1 \) is \( (-4, -4) \). Substitute these coordinates into the common chord equation:
\[ 10(-4) + 4(-4) - (a + b) = 0 \]
Calculating gives:
\[ -40 - 16 - (a + b) = 0 \]
Thus:
\[ -56 = a + b \]

### Step 5: Conclusion
The value of \( a + b \) is:
\[ a + b = -56 \]

### Final Answer:
The answer is \( -56 \), which corresponds to option (C).

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