If the coordinate axes are the bisectors of the angles between the pair of lines `ax^(2)+2hxy+by^(2)=0`, then

Options

A. (a) a=b
B. (b) h=0
C. (c) `a^2=b=0`
D. (d) `a+b^2=0`

Answer

To solve the problem, we need to analyze the condition given in the question: the coordinate axes are the bisectors of the angles between the pair of lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \).

### Step-by-Step Solution:

1. **Understand the Given Equation**:
The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines. The coefficients \( a \), \( b \), and \( h \) determine the nature of these lines.

2. **Equation of Angle Bisectors**:
The angle bisectors of the lines represented by the equation can be expressed using the formula:
\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{h} \]
This is derived from the general form of the angle bisectors of the lines.

3. **Cross Multiply**:
Rearranging the angle bisector equation gives:
\[ h(x^2 - y^2) = (a - b)xy \]
We will refer to this as Equation (1).

4. **Substituting the Condition**:
Since the coordinate axes (i.e., \( x = 0 \) and \( y = 0 \)) are the angle bisectors, we can substitute \( xy = 0 \) into Equation (1).

5. **Evaluate the Substitution**:
Substituting \( xy = 0 \) into Equation (1) results in:
\[ h(x^2 - y^2) = 0 \]
This implies that either:
- \( h = 0 \) or
- \( x^2 - y^2 = 0 \) (which means \( x = y \) or \( x = -y \))

6. **Analyzing the Conditions**:
If \( x^2 - y^2 = 0 \), then it does not provide a definitive condition on \( a \) and \( b \) since \( a - b \) cannot be zero for the equation to hold true. Therefore, we must have \( h = 0 \).

7. **Conclusion**:
The only valid condition that satisfies the requirement that the coordinate axes are the angle bisectors is:
\[ h = 0 \]
Thus, the correct answer is option (b): \( h = 0 \).