The straight lines `x+2y-9=0,3x+5y-5=0` , and `a x+b y-1=0` are concurrent, if the straight line `35 x-22 y+1=0` passes through the point `(a , b)` (b) `(b ,a)` `(-a ,-b)` (d) none of these

Options

A. `(a,b)`
B. `(b,a)`
C. `(a,-b)`
D. `(-a,b)`

Answer

To determine the point \((a, b)\) such that the lines \(x + 2y - 9 = 0\), \(3x + 5y - 5 = 0\), and \(ax + by - 1 = 0\) are concurrent, and the line \(35x - 22y + 1 = 0\) passes through this point, we can follow these steps:

### Step 1: Set up the condition for concurrency
The three lines are concurrent if the determinant of their coefficients is zero. Therefore, we can write the determinant as follows:

\[ \begin{vmatrix} 1 & 2 & -9 \\ 3 & 5 & -5 \\ a & b & -1 \end{vmatrix} = 0 \]

### Step 2: Calculate the determinant
Calculating the determinant, we have:

\[ 1 \cdot (5 \cdot (-1) - (-5) \cdot b) - 2 \cdot (3 \cdot (-1) - (-5) \cdot a) - 9 \cdot (3b - 5a) = 0 \]

This simplifies to:

\[ 1 \cdot (-5 + 5b) - 2 \cdot (-3 + 5a) - 9 \cdot (3b - 5a) = 0 \]

### Step 3: Expand and simplify the equation
Expanding this, we get:

\[ -5 + 5b + 6 - 10a - 27b + 45a = 0 \]

Combining like terms results in:

\[ (35a - 22b + 1) = 0 \]

### Step 4: Set up the equation with the line \(35x - 22y + 1 = 0\)
Since the line \(35x - 22y + 1 = 0\) passes through the point \((a, b)\), we can substitute \(x = a\) and \(y = b\):

\[ 35a - 22b + 1 = 0 \]

### Step 5: Solve for \(a\) and \(b\)
From the equation \(35a - 22b + 1 = 0\), we can express \(b\) in terms of \(a\):

\[ 22b = 35a + 1 \implies b = \frac{35a + 1}{22} \]

### Step 6: Identify the point \((a, b)\)
Now we have \(b\) expressed in terms of \(a\). Thus, the point \((a, b)\) can be represented as:

\[ \left(a, \frac{35a + 1}{22}\right) \]

### Step 7: Check the options
We need to check which of the given options corresponds to this point. The options provided were:

(a) \((a, b)\)
(b) \((b, a)\)
(c) \((-a, -b)\)
(d) none of these

Since we have \(b = \frac{35a + 1}{22}\), option (a) \((a, b)\) is indeed the correct representation of the point.

### Conclusion
The correct answer is (a) \((a, b)\).

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