If the circle `x^2+y^2-6x-4y+9=0` bisects the circumference of the circle `x^2+y^2-8x-6y+a=0` , then the value of a is ____

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the centers of the circles
The equations of the circles are given as:
1. First circle: \( x^2 + y^2 - 6x - 4y + 9 = 0 \)
2. Second circle: \( x^2 + y^2 - 8x - 6y + a = 0 \)

To find the center of the first circle, we convert it to the standard form. The center \((h, k)\) of a circle given by the equation \(x^2 + y^2 + Dx + Ey + F = 0\) is given by:
\[ (h, k) = \left(-\frac{D}{2}, -\frac{E}{2}\right) \]
For the first circle:
- \(D = -6\)
- \(E = -4\)

Thus, the center \(C_1\) is:
\[ C_1 = \left(-\frac{-6}{2}, -\frac{-4}{2}\right) = (3, 2) \]

### Step 2: Find the center of the second circle
For the second circle:
- \(D = -8\)
- \(E = -6\)

Thus, the center \(C_2\) is:
\[ C_2 = \left(-\frac{-8}{2}, -\frac{-6}{2}\right) = (4, 3) \]

### Step 3: Use the condition of bisection
Since the first circle bisects the circumference of the second circle, the common chord of both circles is the diameter of the second circle. The equation of the common chord can be derived from the equations of both circles.

### Step 4: Derive the equation of the common chord
The equation of the common chord can be found using:
\[ S_1 - S_2 = 0 \]
where \(S_1\) and \(S_2\) are the equations of the first and second circles respectively.

Substituting the equations:
\[ S_1: x^2 + y^2 - 6x - 4y + 9 = 0 \]
\[ S_2: -x^2 - y^2 + 8x + 6y - a = 0 \]

Now, we can write:
\[ S_1 - S_2: (x^2 + y^2 - 6x - 4y + 9) - (-x^2 - y^2 + 8x + 6y - a) = 0 \]
This simplifies to:
\[ 2x + 2y + 9 - a = 0 \]

### Step 5: Substitute the center of the second circle
Since the common chord passes through the center of the second circle \(C_2(4, 3)\), we substitute \(x = 4\) and \(y = 3\) into the equation:
\[ 2(4) + 2(3) + 9 - a = 0 \]
Calculating this gives:
\[ 8 + 6 + 9 - a = 0 \]
\[ 23 - a = 0 \]

### Step 6: Solve for \(a\)
Rearranging gives:
\[ a = 23 \]

Thus, the value of \(a\) is \(23\).

### Final Answer:
The value of \(a\) is **23**.

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