The bisectors of the angles between the lines `(ax+by)^2=c(bx-ay)^2,c gt0` are respectively parallel and perpendicular to the line

Options

A. `bx-ay+mu=0`
B. `ax+by+lambda=0`
C. `ax=by+v=0`
D. `bx+ay+tau=0`

Answer

To solve the problem, we need to analyze the given equations and find the conditions under which the bisectors of the angles between the lines are parallel and perpendicular to a given line.

### Step-by-Step Solution:

1. **Understanding the Given Lines**:
The lines given are:
\[ (ax + by)^2 = c(bx - ay)^2 \]
where \( c > 0 \).

2. **Expanding the Equation**:
We can expand both sides of the equation:
\[ (ax + by)^2 = a^2x^2 + b^2y^2 + 2abxy \]
\[ c(bx - ay)^2 = c(b^2x^2 + a^2y^2 - 2abxy) \]
Setting these equal gives:
\[ a^2x^2 + b^2y^2 + 2abxy = c(b^2x^2 + a^2y^2 - 2abxy) \]

3. **Rearranging the Equation**:
Rearranging the equation leads to:
\[ a^2x^2 + b^2y^2 + 2abxy - cb^2x^2 - ca^2y^2 + 2cabxy = 0 \]
This simplifies to:
\[ (a^2 - cb^2)x^2 + (b^2 - ca^2)y^2 + (2ab + 2cab)xy = 0 \]

4. **Identifying the Coefficients**:
Let:
\[ A = a^2 - cb^2, \quad B = b^2 - ca^2, \quad H = 2ab + 2cab \]

5. **Equation of the Angle Bisectors**:
The equation of the angle bisectors can be expressed as:
\[ \frac{x^2 - y^2}{xy} = \frac{A}{H} \quad \text{or} \quad \frac{x^2 - y^2}{xy} = \frac{B}{H} \]

6. **Finding the Slopes of the Bisectors**:
Let \( m_1 \) and \( m_2 \) be the slopes of the angle bisectors. The sum and product of the slopes can be given by:
\[ m_1 + m_2 = -\frac{2H}{B}, \quad m_1 m_2 = -1 \]

7. **Condition for Parallel and Perpendicular**:
Given that one bisector is parallel to the line \( bx - ay + \mu = 0 \) and the other is perpendicular, we can set the slopes accordingly:
- If the slope of the line is \( -\frac{b}{a} \), then one bisector must equal this slope.
- The other bisector's slope must be the negative reciprocal, \( \frac{a}{b} \).

8. **Conclusion**:
From the conditions derived, we can conclude that the correct option must satisfy the relationship between the slopes derived from the angle bisectors and the given line.

### Final Answer:
The correct option is:
\[ \text{Option B: } ax + by + \lambda = 0 \]