Statement -1 : The lines (a + b) x + 2(a - b) y = 2a are concurrent at the point (1, 1/2) .
Statement-2 : `L_(1) + lambda L_(2) = 0` represents the equation of family of lines passing through the intersection of the lines `L_(1) = 0` and `L_(2) = 0` for all non - zero finite value of `lambda`

Options

A. Statement -1 is True , Statement - 2 is true , Statement- 2 is a correct explanation for statement - 1
B. Statement-1 is True , Statement-2 is True , Statement -2 is not a correct explanation for Statement - 1 .
C. Statement-1 is True , Statement - 2 is False .
D. Statement - 1 is False , Statement -2 is True .

Explanation

To solve the problem, we need to analyze both statements and verify their correctness step by step.

### Step 1: Analyze Statement 1
The first statement claims that the lines given by the equation \((a + b)x + 2(a - b)y = 2a\) are concurrent at the point \((1, \frac{1}{2})\).

**Verification:**
1. Substitute \(x = 1\) and \(y = \frac{1}{2}\) into the equation:
\[ (a + b)(1) + 2(a - b)\left(\frac{1}{2}\right) = 2a \]
Simplifying this gives:
\[ a + b + (a - b) = 2a \]
\[ 2a = 2a \]
This is true for all values of \(a\) and \(b\). Thus, the point \((1, \frac{1}{2})\) lies on the line.

### Step 2: Analyze Statement 2
The second statement states that \(L_1 + \lambda L_2 = 0\) represents the equation of a family of lines passing through the intersection of the lines \(L_1 = 0\) and \(L_2 = 0\) for all non-zero finite values of \(\lambda\).

**Verification:**
1. If \(L_1 = 0\) and \(L_2 = 0\) are two lines, the equation \(L_1 + \lambda L_2 = 0\) represents a linear combination of these two lines.
2. The intersection of \(L_1\) and \(L_2\) gives a point. The equation \(L_1 + \lambda L_2 = 0\) indeed represents a family of lines that pass through this intersection point for any non-zero \(\lambda\).

### Conclusion
Both statements are true. Statement 2 correctly explains Statement 1.

### Final Answer
Both Statement 1 and Statement 2 are true, with Statement 2 being the correct explanation for Statement 1.

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