To find the condition that the three straight lines `A_(1)x+B_(1)y+C_(1)=0, A_(2)x+B_(2)y+C_(2)=0 and A_(3)x+B_(3)y+C_(3)=0` are concurrent.

Answer

To find the condition that the three straight lines \( A_1x + B_1y + C_1 = 0 \), \( A_2x + B_2y + C_2 = 0 \), and \( A_3x + B_3y + C_3 = 0 \) are concurrent, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Concurrent Lines**:
Three lines are said to be concurrent if they all intersect at a single point. This means that there exists a common solution (x, y) that satisfies all three equations simultaneously.

2. **Setting Up the Determinant**:
To find the condition for concurrency, we can use the determinant of a matrix formed by the coefficients of the lines. The lines can be represented in the form:
\[ \begin{align*} A_1x + B_1y + C_1 &= 0 \\ A_2x + B_2y + C_2 &= 0 \\ A_3x + B_3y + C_3 &= 0 \end{align*} \]
We can form a matrix using the coefficients of \(x\), \(y\), and the constant terms:
\[ \begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{vmatrix} \]

3. **Condition for Concurrency**:
The condition for the three lines to be concurrent is that the determinant of this matrix must equal zero:
\[ \begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{vmatrix} = 0 \]

4. **Expanding the Determinant**:
The determinant can be expanded as follows:
\[ A_1(B_2C_3 - B_3C_2) - B_1(A_2C_3 - A_3C_2) + C_1(A_2B_3 - A_3B_2) = 0 \]
This equation represents the condition for concurrency.

### Final Condition:
Thus, the condition that the three lines \( A_1x + B_1y + C_1 = 0 \), \( A_2x + B_2y + C_2 = 0 \), and \( A_3x + B_3y + C_3 = 0 \) are concurrent is given by:
\[ \begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{vmatrix} = 0 \]