If `6a^2-3b^2-c^2+7ab-ac+4bc=0` then the family of lines `ax+by+c=0,|a|+|b| != 0` can be concurrent at concurrent

Options

A. `(-2,-3)`
B. `(3,-1)`
C. `(2,3)`
D. `(-3,1)`

Answer

To solve the problem, we need to analyze the given equation and determine the conditions under which the family of lines represented by \( ax + by + c = 0 \) can be concurrent.

### Step-by-Step Solution:

1. **Given Equation**:
We start with the equation:
\[ 6a^2 - 3b^2 - c^2 + 7ab - ac + 4bc = 0 \]

2. **Rearranging the Equation**:
We can rearrange the equation to group similar terms:
\[ 6a^2 + 7ab - ac - 3b^2 + 4bc - c^2 = 0 \]

3. **Finding the Condition for Concurrency**:
For the lines \( ax + by + c = 0 \) to be concurrent, the determinant formed by the coefficients must be zero. The determinant condition can be derived from the general form of the lines:
\[ \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0 \]
In our case, we need to find the values of \( a, b, c \) that satisfy the given equation.

4. **Substituting Values**:
We can substitute different values of \( a, b, c \) from the options provided to check which ones satisfy the equation. The options are:
- Option 1: (-2, -3)
- Option 2: (3, -1)
- Option 3: (2, 3)
- Option 4: (-3, 1)

5. **Testing the Options**:
We will substitute each pair into the equation to see if it holds true.

- **Option 1: \( a = -2, b = -3 \)**:
\[ 6(-2)^2 - 3(-3)^2 - c^2 + 7(-2)(-3) - (-2)c + 4(-3)c = 0 \]
Simplifying gives:
\[ 24 - 27 - c^2 + 42 + 2c - 12c = 0 \]
\[ -c^2 - 10c + 39 = 0 \]
This is a quadratic in \( c \) which can have real solutions.

- **Option 2: \( a = 3, b = -1 \)**:
\[ 6(3)^2 - 3(-1)^2 - c^2 + 7(3)(-1) - 3c + 4(-1)c = 0 \]
Simplifying gives:
\[ 54 - 3 - c^2 - 21 - 3c - 4c = 0 \]
\[ -c^2 - 7c + 30 = 0 \]
This also has real solutions.

- **Option 3: \( a = 2, b = 3 \)**:
\[ 6(2)^2 - 3(3)^2 - c^2 + 7(2)(3) - 2c + 4(3)c = 0 \]
Simplifying gives:
\[ 24 - 27 - c^2 + 42 - 2c + 12c = 0 \]
\[ -c^2 + 10c + 39 = 0 \]
This also has real solutions.

- **Option 4: \( a = -3, b = 1 \)**:
\[ 6(-3)^2 - 3(1)^2 - c^2 + 7(-3)(1) - (-3)c + 4(1)c = 0 \]
Simplifying gives:
\[ 54 - 3 - c^2 - 21 + 3c + 4c = 0 \]
\[ -c^2 + 7c + 30 = 0 \]
This also has real solutions.

6. **Conclusion**:
The pairs that satisfy the concurrency condition are found in options 1 and 2. Thus, the correct options are:
- Option 1: (-2, -3)
- Option 2: (3, -1)