The circle `x^(2)+y^(2)=4` cuts the circle `x^(2)+y^(2)-2x-4=0` at the points A and B. If the circle `x^(2)+y^(2)-4x-k=0` passes through A and B then the value of k , is

Options

A. -4
B. 0
C. -8
D. 4

Answer

To solve the problem, we need to find the value of \( k \) such that the circle \( x^2 + y^2 - 4x - k = 0 \) passes through the points of intersection \( A \) and \( B \) of the circles \( x^2 + y^2 = 4 \) and \( x^2 + y^2 - 2x - 4 = 0 \).

### Step 1: Identify the equations of the circles
The first circle is given by:
\[ S_1: x^2 + y^2 = 4 \]
The second circle can be rewritten as:
\[ S_2: x^2 + y^2 - 2x - 4 = 0 \implies x^2 + y^2 = 2x + 4 \]

### Step 2: Use the family of circles concept
According to the family of circles concept, if two circles \( S_1 \) and \( S_2 \) intersect at points \( A \) and \( B \), then any circle passing through \( A \) and \( B \) can be expressed as:
\[ S_3: S_1 + \lambda S_2 = 0 \]
for some parameter \( \lambda \).

### Step 3: Write the equation of the new circle
Substituting \( S_1 \) and \( S_2 \) into the equation for \( S_3 \):
\[ S_3: (x^2 + y^2 - 4) + \lambda (x^2 + y^2 - 2x - 4) = 0 \]
This simplifies to:
\[ (1 + \lambda)x^2 + (1 + \lambda)y^2 - 2\lambda x - (4 + 4\lambda) = 0 \]

### Step 4: Rearranging the equation
We can rearrange this equation as:
\[ (1 + \lambda)(x^2 + y^2) - 2\lambda x - (4 + 4\lambda) = 0 \]

### Step 5: Compare with the given circle equation
The circle we want to compare with is:
\[ x^2 + y^2 - 4x - k = 0 \]
This can be rearranged to:
\[ 1(x^2 + y^2) - 4x - k = 0 \]

### Step 6: Equate coefficients
From the equations, we can compare coefficients:
- Coefficient of \( x^2 + y^2 \): \( 1 + \lambda = 1 \) implies \( \lambda = 0 \)
- Coefficient of \( x \): \( -2\lambda = -4 \) implies \( \lambda = 2 \)
- Constant term: \( -(4 + 4\lambda) = -k \)

### Step 7: Solve for \( k \)
Substituting \( \lambda = 2 \) into the constant term equation:
\[ -(4 + 4 \cdot 2) = -k \implies -12 = -k \implies k = 12 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{12} \]