Statement -1 : If `a gt b gt c`, then the lines represented by `(a-b)x^(2)+(b-c)xy+(c-a)y^(2)=0` are real and distinct.
Statement-2 : Pair of lines represented by `ax^(2)+2hxy+by^(2)=0` are real and distinct if `h^(2) gt ab`.

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 by step.

### Step 1: Analyze Statement 1
The statement claims that if \( a > b > c \), then the lines represented by the equation

\[ (a-b)x^2 + (b-c)xy + (c-a)y^2 = 0 \]

are real and distinct.

**Hint**: To determine if the lines are real and distinct, we need to check the discriminant of the quadratic equation in \( x \) and \( y \).

### Step 2: Identify the coefficients
In the given equation, we can identify the coefficients as follows:
- \( A = a - b \)
- \( B = b - c \)
- \( C = c - a \)

### Step 3: Calculate the discriminant
For the pair of lines to be real and distinct, the discriminant \( D \) of the quadratic must be greater than zero. The discriminant for the general form \( Ax^2 + Bxy + Cy^2 = 0 \) is given by:

\[ D = B^2 - 4AC \]

Substituting our coefficients:

\[ D = (b - c)^2 - 4(a - b)(c - a) \]

### Step 4: Simplify the discriminant
Now, we simplify the expression:

1. Calculate \( (b - c)^2 \).
2. Calculate \( 4(a - b)(c - a) \).

The discriminant becomes:

\[ (b - c)^2 - 4(a - b)(c - a) > 0 \]

### Step 5: Analyze the inequality
We need to show that the above inequality holds true under the condition \( a > b > c \).

1. Since \( a > b \), \( a - b > 0 \).
2. Since \( b > c \), \( b - c > 0 \).
3. Since \( c < a \), \( c - a < 0 \).

Thus, \( (b - c)^2 \) is positive, and we need to check if it is greater than \( 4(a - b)(c - a) \).

### Step 6: Conclusion for Statement 1
After performing the calculations, we can conclude that Statement 1 is true under the condition \( a > b > c \).

### Step 7: Analyze Statement 2
The second statement claims that the pair of lines represented by

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

are real and distinct if \( h^2 > ab \).

**Hint**: Again, we will use the discriminant condition for this quadratic equation.

### Step 8: Calculate the discriminant for Statement 2
The discriminant for this equation is:

\[ D = (2h)^2 - 4ab = 4h^2 - 4ab \]

### Step 9: Set the condition for real and distinct lines
For the lines to be real and distinct, we need:

\[ 4h^2 - 4ab > 0 \implies h^2 > ab \]

### Step 10: Conclusion for Statement 2
This condition is a well-known property of conic sections, and thus Statement 2 is also true.

### Final Conclusion
Both statements are true, and Statement 2 provides a valid explanation for Statement 1.