If the straight lines `x+2y-9=0, 3x+5y-5=0` and `ax+by-1=0` are concurrent, then the straight line `35x-22y-1=0` passes through

Options

A. (a,-b)
B. (a,b)
C. (-a,b)
D. None

Answer

To solve the problem, we need to determine the values of \(a\) and \(b\) such that the lines \(x + 2y - 9 = 0\), \(3x + 5y - 5 = 0\), and \(ax + by - 1 = 0\) are concurrent. We will then check if the line \(35x - 22y - 1 = 0\) passes through the point of concurrency.

### Step-by-Step Solution:

1. **Identify the lines**:
We have three lines:
- Line 1: \(x + 2y - 9 = 0\) (Coefficients: \(a_1 = 1, b_1 = 2, c_1 = -9\))
- Line 2: \(3x + 5y - 5 = 0\) (Coefficients: \(a_2 = 3, b_2 = 5, c_2 = -5\))
- Line 3: \(ax + by - 1 = 0\) (Coefficients: \(a_3 = a, b_3 = b, c_3 = -1\))

2. **Set up the determinant for concurrency**:
For the three lines to be concurrent, the determinant must be zero:
\[ \Delta = \begin{vmatrix} 1 & 2 & -9 \\ 3 & 5 & -5 \\ a & b & -1 \end{vmatrix} = 0 \]

3. **Calculate the determinant**:
Expanding the determinant along the first row:
\[ \Delta = 1 \cdot \begin{vmatrix} 5 & -5 \\ b & -1 \end{vmatrix} - 2 \cdot \begin{vmatrix} 3 & -5 \\ a & -1 \end{vmatrix} - 9 \cdot \begin{vmatrix} 3 & 5 \\ a & b \end{vmatrix} \]

Calculating each of these 2x2 determinants:
- \(\begin{vmatrix} 5 & -5 \\ b & -1 \end{vmatrix} = 5(-1) - (-5)b = -5 + 5b = 5b - 5\)
- \(\begin{vmatrix} 3 & -5 \\ a & -1 \end{vmatrix} = 3(-1) - (-5)a = -3 + 5a = 5a - 3\)
- \(\begin{vmatrix} 3 & 5 \\ a & b \end{vmatrix} = 3b - 5a\)

Plugging these back into the determinant:
\[ \Delta = 1(5b - 5) - 2(5a - 3) - 9(3b - 5a) = 5b - 5 - 10a + 6 - 27b + 45a \]
Simplifying:
\[ \Delta = (5b - 27b) + (-10a + 45a) + (6 - 5) = -22b + 35a + 1 \]

4. **Set the determinant to zero**:
\[ -22b + 35a + 1 = 0 \]
Rearranging gives:
\[ 35a - 22b + 1 = 0 \]

5. **Compare with the line \(35x - 22y - 1 = 0\)**:
We can rewrite \(35a - 22b + 1 = 0\) as:
\[ 35a - 22b = -1 \]
This shows that if we set \(x = -a\) and \(y = -b\), the line \(35x - 22y - 1 = 0\) will pass through the point \((-a, -b)\).

### Conclusion:
Thus, the line \(35x - 22y - 1 = 0\) passes through the point \((-a, -b)\).