If the pair of straight lines `ax^(2)+2hxy-ay^(2)=0` and `bx^(2)+2gxy-by^(2)=0` be such that each bisects the angle between the other then

Options

A. `ag+bh=0`
B. `bg+ah=0`
C. `gh+ab=0`
D. `h^(2)-ab=0`

Answer

To solve the problem, we need to analyze the given equations of the pair of straight lines and derive the condition under which each line bisects the angle between the other.

### Step-by-Step Solution:

1. **Understand the Given Equations**:
The equations of the two pairs of straight lines are:
\[ ax^2 + 2hxy - ay^2 = 0 \quad \text{(1)} \]
\[ bx^2 + 2gxy - by^2 = 0 \quad \text{(2)} \]

2. **Angle Bisector Condition**:
The condition for the angle bisectors of two pairs of lines is given by:
\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{h} \quad \text{(for the first line)} \]
and
\[ \frac{x^2 - y^2}{b - a} = \frac{xy}{g} \quad \text{(for the second line)} \]

3. **Deriving the Angle Bisector Equation**:
From equation (1), we can express the angle bisector:
\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{h} \implies h(x^2 - y^2) = (a - b)xy \quad \text{(3)} \]

4. **Substituting into the Second Equation**:
Now, we substitute the angle bisector condition from (3) into the second equation (2):
\[ b(x^2 - y^2) + 2gxy - b(y^2) = 0 \]
This gives us:
\[ b(x^2 - y^2) + 2gxy = 0 \quad \text{(4)} \]

5. **Comparing Coefficients**:
From equation (3) and the derived equation (4), we can compare coefficients:
\[ \frac{h}{b} = \frac{-2a}{2g} \implies hg = -ab \quad \text{(5)} \]

6. **Final Result**:
Rearranging equation (5) gives us the required relation:
\[ hg + ab = 0 \]

### Conclusion:
The condition under which each line bisects the angle between the other is:
\[ hg + ab = 0 \]