The centre of family of circles cutting the family of circles `x^2 + y^2 + 4x(lambda-3/2) + 3y(lambda-4/3)-6(lambda+2)=0` orthogonally, lies on

Options

A. `x - y - 1 = 0`
B. `4x +3y - 6 = 0`
C. `4x +3y +7 = 0`
D. `3x - 4y - 1 = 0`

Explanation

To solve the problem, we need to find the equation of the line on which the centers of the family of circles that cut the given family of circles orthogonally lie.

### Step-by-Step Solution:

1. **Understand the Given Circle Equation**:
The given family of circles is represented by the equation:
\[ x^2 + y^2 + 4x\left(\lambda - \frac{3}{2}\right) + 3y\left(\lambda - \frac{4}{3}\right) - 6(\lambda + 2) = 0 \]
This can be rewritten in a more manageable form by grouping terms involving \(\lambda\).

2. **Rearranging the Circle Equation**:
Rearranging the terms, we have:
\[ x^2 + y^2 + 4x\lambda - 6x + 3y\lambda - 4y - 6\lambda + 12 = 0 \]
This separates the \(\lambda\) dependent and independent terms.

3. **Identifying \(s_1\) and \(s_2\)**:
We can express the equation in terms of \(s_1\) and \(s_2\) for two different values of \(\lambda\):
\[ s_1 = x^2 + y^2 - 6x - 4y - 12 + \lambda_1(4x - 6y - 6) = 0 \]
\[ s_2 = x^2 + y^2 - 6x - 4y - 12 + \lambda_2(4x - 6y - 6) = 0 \]

4. **Finding the Radical Line**:
The radical line is determined by the equation \(s_1 - s_2 = 0\):
\[ (s_1 - s_2) = \lambda_1(4x - 6y - 6) - \lambda_2(4x - 6y - 6) = 0 \]
This simplifies to:
\[ (4x - 6y - 6)(\lambda_1 - \lambda_2) = 0 \]

5. **Conclusion**:
Since \(\lambda_1 \neq \lambda_2\), we can conclude that:
\[ 4x - 6y - 6 = 0 \]
This is the equation of the radical line on which the centers of the family of circles lie.

### Final Answer:
The equation of the line on which the center of the family of circles lies is:
\[ 4x - 6y - 6 = 0 \]