If the lines `x - y + 1 = 0, 2x + 3y + 2 = 0 and ax + by - 2 = 0` are concurrent, then `5x + 10 = 0` passes through the point

Options

A. (a, b)
B. (b, a)
C. (0, b)
D. (a, 0)

Answer

To solve the problem, we need to determine the conditions under which the lines \( x - y + 1 = 0 \), \( 2x + 3y + 2 = 0 \), and \( ax + by - 2 = 0 \) are concurrent. This means that they all intersect at a single point.

### Step 1: Find the intersection of the first two lines

We have the equations of the first two lines:
1. \( x - y + 1 = 0 \) (Let's call this Line 1)
2. \( 2x + 3y + 2 = 0 \) (Let's call this Line 2)

We can rewrite Line 1 in terms of \( y \):
\[ y = x + 1 \]

Now, substitute \( y \) from Line 1 into Line 2:
\[ 2x + 3(x + 1) + 2 = 0 \]
\[ 2x + 3x + 3 + 2 = 0 \]
\[ 5x + 5 = 0 \]
\[ 5x = -5 \]
\[ x = -1 \]

Now, substitute \( x = -1 \) back into the equation of Line 1 to find \( y \):
\[ y = -1 + 1 = 0 \]

Thus, the point of intersection of the first two lines is:
\[ (-1, 0) \]

### Step 2: Determine the condition for concurrency with the third line

The third line is given by:
\[ ax + by - 2 = 0 \]

For the three lines to be concurrent, the point \( (-1, 0) \) must satisfy this equation. Substitute \( x = -1 \) and \( y = 0 \) into the equation:
\[ a(-1) + b(0) - 2 = 0 \]
\[ -a - 2 = 0 \]
\[ -a = 2 \]
\[ a = -2 \]

### Step 3: Analyze the value of \( b \)

Since there are no restrictions on \( b \), it can take any real value. Therefore, \( b \) can be any real number.

### Step 4: Check the line \( 5x + 10 = 0 \)

Now, we need to check if the line \( 5x + 10 = 0 \) passes through the point \( (-1, 0) \):
\[ 5x + 10 = 0 \Rightarrow 5(-1) + 10 = -5 + 10 = 5 \neq 0 \]

This means that the line \( 5x + 10 = 0 \) does not pass through the point \( (-1, 0) \).

### Conclusion

The lines \( x - y + 1 = 0 \), \( 2x + 3y + 2 = 0 \), and \( ax + by - 2 = 0 \) are concurrent when \( a = -2 \) and \( b \) can be any real number. However, the line \( 5x + 10 = 0 \) does not pass through the point of intersection \( (-1, 0) \).