One of the lines given by the equation `ax^2+2lamdaxy+by^2=0` will bisect the angle between the coordinate axes if

Options

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

Answer

To determine the condition under which one of the lines given by the equation \( ax^2 + 2\lambda axy + by^2 = 0 \) bisects the angle between the coordinate axes, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Geometry**: The lines that bisect the angle between the coordinate axes (x-axis and y-axis) will have slopes of \( m = 1 \) or \( m = -1 \). This is because the angle between the axes is \( 90^\circ \), and the bisectors of this angle will be at \( 45^\circ \) and \( -45^\circ \).

2. **Rewrite the Equation**: The given equation is \( ax^2 + 2\lambda axy + by^2 = 0 \). We can rewrite this in terms of \( y/x \) (let \( m = y/x \)):
\[ a + 2\lambda m + bm^2 = 0 \]

3. **Substitute the Slopes**: Since we want the lines to have slopes \( m = 1 \) and \( m = -1 \), we will substitute these values into the equation.

- For \( m = 1 \):
\[ a + 2\lambda(1) + b(1^2) = 0 \implies a + 2\lambda + b = 0 \]

- For \( m = -1 \):
\[ a + 2\lambda(-1) + b(-1^2) = 0 \implies a - 2\lambda + b = 0 \]

4. **Set Up the System of Equations**: We now have two equations:
\[ 1) \quad a + 2\lambda + b = 0 \]
\[ 2) \quad a - 2\lambda + b = 0 \]

5. **Solve the System**: Subtract the second equation from the first:
\[ (a + 2\lambda + b) - (a - 2\lambda + b) = 0 \implies 4\lambda = 0 \implies \lambda = 0 \]

6. **Substitute Back**: If \( \lambda = 0 \), substitute it back into either equation:
\[ a + b = 0 \implies b = -a \]

7. **Conclusion**: Therefore, one of the lines given by the equation will bisect the angle between the coordinate axes if:
\[ \lambda = 0 \quad \text{and} \quad b = -a \]