STATEMENT-1 : If lines `y = m_(1)x` and `y = m_(2)x` are the conjugate diameter of the hyperbola `xy = c^(2)` then `m_(1) + m_(2) = 0`.
and
STATEMENT-2 : Two lines are called conjugate diameter of hyperbola if they bisect the chords parallel to each other.

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

Answer

To solve the problem, we need to show that if the lines \( y = m_1 x \) and \( y = m_2 x \) are conjugate diameters of the hyperbola \( xy = c^2 \), then \( m_1 + m_2 = 0 \).

### Step-by-Step Solution:

1. **Understanding Conjugate Diameters**:
Conjugate diameters of a hyperbola are defined as two lines that bisect chords that are parallel to each other. For the hyperbola \( xy = c^2 \), we need to establish the relationship between the slopes of these lines.

**Hint**: Recall the definition of conjugate diameters in the context of hyperbolas.

2. **Equation of the Hyperbola**:
The given hyperbola is \( xy = c^2 \). This is a rectangular hyperbola, which means that the axes are symmetric.

**Hint**: Identify the properties of rectangular hyperbolas.

3. **Setting Up the Lines**:
We have two lines:
\[ y = m_1 x \quad \text{and} \quad y = m_2 x \]
If these lines are conjugate diameters, they must bisect chords of the hyperbola that are parallel to each other.

**Hint**: Think about how the slopes \( m_1 \) and \( m_2 \) relate to the geometry of the hyperbola.

4. **Finding the Relationship Between Slopes**:
For the lines to be conjugate diameters, the slopes must satisfy the condition:
\[ m_2 = -m_1 \]
This is because the lines must be symmetric about the origin due to the nature of the rectangular hyperbola.

**Hint**: Use symmetry properties of the hyperbola to derive the relationship between the slopes.

5. **Adding the Slopes**:
Now, if we add \( m_1 \) and \( m_2 \):
\[ m_1 + m_2 = m_1 + (-m_1) = 0 \]

**Hint**: Simplifying the expression will show the relationship clearly.

6. **Conclusion**:
Therefore, we have shown that if \( y = m_1 x \) and \( y = m_2 x \) are conjugate diameters of the hyperbola \( xy = c^2 \), then indeed \( m_1 + m_2 = 0 \).

**Hint**: Reflect on how this conclusion aligns with the definitions and properties discussed.

### Final Answer:
Both statements are true:
- Statement 1 is correct: \( m_1 + m_2 = 0 \).
- Statement 2 correctly explains the concept of conjugate diameters.