If one of the lines of the pair `a x^2+2h x y+b y^2=0` bisects the angle between the positive direction of the axes. Then find the relation for `a ,b ,a n dhdot`

Explanation

To solve the problem, we need to find the relationship between the coefficients \( a \), \( b \), and \( h \) given that one of the lines of the pair \( ax^2 + 2hxy + by^2 = 0 \) bisects the angle between the positive direction of the axes.

### Step-by-step Solution:

1. **Identify the angle bisector**:
The line that bisects the angle between the positive direction of the x-axis and the positive direction of the y-axis is given by the equation \( y = x \).

2. **Substitute the angle bisector into the equation**:
We can substitute \( y = x \) into the equation of the pair of lines:
\[ ax^2 + 2hx \cdot x + bx^2 = 0 \]
This simplifies to:
\[ (a + 2h + b)x^2 = 0 \]

3. **Set the coefficient of \( x^2 \) to zero**:
For the equation to hold true for all \( x \), the coefficient of \( x^2 \) must be zero:
\[ a + 2h + b = 0 \]

4. **Rearrange the equation**:
We can rearrange the equation to express \( h \) in terms of \( a \) and \( b \):
\[ 2h = - (a + b) \]
Thus, we have:
\[ h = -\frac{a + b}{2} \]

5. **Final relationship**:
The relationship between \( a \), \( b \), and \( h \) can be expressed as:
\[ a + b + 2h = 0 \]

### Conclusion:
The required relationship for \( a \), \( b \), and \( h \) is:
\[ a + b + 2h = 0 \]