The two sets of pair of lines given by`ax^(2)-2xy+by^(2)=0` and `bx^(2)-2xy+ay^(2)=0` be such that each pair bisects the angle between the other pair,

Options

A. `a-b=2`
B. `a+b=0`
C. `a+b=2`
D. `a-b=-2`

Answer

To solve the problem, we need to analyze the given equations of pairs of lines and establish the conditions under which each pair bisects the angle between the other pair.

### Step-by-Step Solution:

1. **Write the Given Equations**:
The equations of the pairs of lines are:
\[ ax^2 - 2xy + by^2 = 0 \quad (1) \]
\[ bx^2 - 2xy + ay^2 = 0 \quad (2) \]

2. **Condition for Angle Bisectors**:
For the pairs of lines to bisect each other, the following condition must hold:
\[ \frac{a}{b} = \frac{b}{a} \]
This implies that \( a^2 = b^2 \).

3. **Solve for \( a \) and \( b \)**:
From \( a^2 = b^2 \), we can conclude that:
\[ a = b \quad \text{or} \quad a = -b \]

4. **Substituting Values**:
- If \( a = b \), both equations become identical, which does not provide two distinct pairs of lines.
- If \( a = -b \), we can substitute \( b = -a \) into either equation.

5. **Substituting into Equation (1)**:
Substitute \( b = -a \) into equation (1):
\[ ax^2 - 2xy - ay^2 = 0 \]
This simplifies to:
\[ a(x^2 - y^2) - 2xy = 0 \]

6. **Factoring the Equation**:
Factor the equation:
\[ (x - y)(x + y) = \frac{2xy}{a} \]
This indicates that the lines represented by this equation are dependent on the value of \( a \).

7. **Conclusion**:
The conditions \( a + b = 0 \) and \( a - b = 0 \) lead us to conclude that the only valid solution is \( a = -b \). Therefore, the pairs of lines bisect each other when \( a \) and \( b \) are equal in magnitude but opposite in sign.

### Final Answer:
The values of \( a \) and \( b \) must satisfy \( a + b = 0 \) or \( a = -b \).