If the pair of lines `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 , then |a-b| equals to

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the value of |a - b| given the equations of two pairs of lines. Let's break down the solution step by step.

### Step 1: Write down the equations of the pairs of lines
The equations given are:
1. \( ax^2 - 2xy + by^2 = 0 \)
2. \( bx^2 - 2xy + ay^2 = 0 \)

### Step 2: Understand the condition for angle bisectors
We know that if two pairs of lines bisect each other's angles, their angular bisector equations must be equal. The angular bisector of a pair of lines represented by the equation \( Ax^2 + 2Hxy + By^2 = 0 \) is given by:
\[ \frac{x^2 - y^2}{A - B} = \frac{xy}{-H} \]

### Step 3: Find the angular bisector for the first equation
For the first equation \( ax^2 - 2xy + by^2 = 0 \):
- Here, \( A = a \), \( H = -1 \), and \( B = b \).
- The angular bisector equation becomes:
\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{1} \]
This simplifies to:
\[ a x^2 - y^2 = (a - b) xy \]

### Step 4: Find the angular bisector for the second equation
For the second equation \( bx^2 - 2xy + ay^2 = 0 \):
- Here, \( A = b \), \( H = -1 \), and \( B = a \).
- The angular bisector equation becomes:
\[ \frac{x^2 - y^2}{b - a} = \frac{xy}{1} \]
This simplifies to:
\[ b x^2 - y^2 = (b - a) xy \]

### Step 5: Set the angular bisector equations equal
Since both angular bisector equations must be equal, we can set them equal to each other:
\[ a x^2 - y^2 = (a - b) xy \]
\[ b x^2 - y^2 = (b - a) xy \]

### Step 6: Compare coefficients
From the two equations, we can compare coefficients of \( x^2 \), \( y^2 \), and \( xy \):
1. Coefficient of \( x^2 \):
\[ a = (b - a) \implies a + a = b \implies 2a = b \implies b = 2a \]

2. Coefficient of \( y^2 \):
\[ -1 = -1 \quad \text{(This is always true)} \]

3. Coefficient of \( xy \):
\[ -(a - b) = -1 \implies a - b = 1 \]

### Step 7: Substitute \( b = 2a \) into \( a - b = 1 \)
Substituting \( b = 2a \) into \( a - b = 1 \):
\[ a - 2a = 1 \implies -a = 1 \implies a = -1 \]
Then substituting back to find \( b \):
\[ b = 2(-1) = -2 \]

### Step 8: Calculate |a - b|
Now we can find \( |a - b| \):
\[ |a - b| = |-1 - (-2)| = |-1 + 2| = |1| = 1 \]

### Conclusion
Thus, the final answer is:
\[ |a - b| = 1 \]