Statement-1: If a, b are non-zero real number such that `a+b=2`, then `ax^(2)+2xy+by^(2)+2ax+2by=0` represents a pair of straight lines. Statement-2: If a, b are of opposite signs, then `ax^(2)+2xt+by^(2)=0` represents a pair of distinct lines passing through the origin.

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 determine their validity.

### Step 1: Analyze Statement 1
Statement 1 claims that if \( a, b \) are non-zero real numbers such that \( a + b = 2 \), then the equation

\[ ax^2 + 2hxy + by^2 + 2ax + 2by = 0 \]

represents a pair of straight lines.

#### Step 1.1: Identify the coefficients
The general form of the equation of a pair of straight lines is given by:

\[ Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 \]

From the given equation, we can identify:
- \( A = a \)
- \( H = 1 \) (since \( 2h = 2 \))
- \( B = b \)
- \( G = a \)
- \( F = b \)
- \( C = 0 \)

#### Step 1.2: Apply the condition for a pair of straight lines
The condition for the equation to represent a pair of straight lines is given by:

\[ ABC + 2FGH - AF^2 - BG^2 - CH^2 = 0 \]

Substituting the identified coefficients into this condition, we get:

\[ a \cdot b \cdot 0 + 2 \cdot b \cdot a \cdot 1 - a \cdot b^2 - b \cdot a^2 - 0 \cdot 1^2 = 0 \]

This simplifies to:

\[ 2ab - ab^2 - a^2b = 0 \]

#### Step 1.3: Factor the equation
Factoring out \( ab \):

\[ ab(2 - a - b) = 0 \]

Since \( a \) and \( b \) are non-zero, we have:

\[ 2 - a - b = 0 \quad \Rightarrow \quad a + b = 2 \]

Thus, Statement 1 is **true**.

### Step 2: Analyze Statement 2
Statement 2 claims that if \( a, b \) are of opposite signs, then the equation

\[ ax^2 + 2xy + by^2 = 0 \]

represents a pair of distinct lines passing through the origin.

#### Step 2.1: Identify the coefficients
In this case, we have:
- \( A = a \)
- \( H = 1 \)
- \( B = b \)
- \( G = 0 \)
- \( F = 0 \)
- \( C = 0 \)

#### Step 2.2: Apply the condition for distinct lines
For the equation to represent distinct lines, the condition is:

\[ 1 - ab > 0 \]

Since \( a \) and \( b \) are of opposite signs, \( ab < 0 \). Therefore, we have:

\[ 1 - ab > 1 > 0 \]

This means that the condition is satisfied, and the equation represents distinct lines passing through the origin.

Thus, Statement 2 is also **true**.

### Conclusion
Both statements are true, but Statement 2 does not provide a correct explanation for Statement 1. Therefore, the answer is that both statements are true, but Statement 2 is not a correct explanation for Statement 1.