If a pair of lines `x^(2)-2px-y^(2)=0` and `x^(2)-2qxy-y^(2)=0` is such that each pair bisects the angle between the other pair, then

Options

A. `pq=-1`
B. `pq=1`
C. `1/p+1/q=0`
D. `1/p-1/q=0`

Answer

To solve the problem, we need to analyze the given equations of the lines and determine the conditions under which each pair bisects the angle between the other pair.

### Step-by-Step Solution:

1. **Identify the given equations**:
The equations of the lines are:
\[ x^2 - 2px - y^2 = 0 \quad \text{(1)} \]
\[ x^2 - 2qxy - y^2 = 0 \quad \text{(2)} \]

2. **Rewrite the equations in standard form**:
The equations can be rewritten in the standard conic form:
\[ A_1 = 1, \quad B_1 = 0, \quad C_1 = -1, \quad D_1 = -2p \quad \text{(for equation 1)} \]
\[ A_2 = 1, \quad B_2 = -2q, \quad C_2 = -1, \quad D_2 = 0 \quad \text{(for equation 2)} \]

3. **Use the angle bisector condition**:
For the lines to bisect the angles between each other, the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

4. **Set up the equations**:
From the coefficients:
- From \(A_1\) and \(A_2\):
\[ \frac{1}{1} = 1 \]
- From \(B_1\) and \(B_2\):
\[ \frac{-2p}{-2q} = \frac{p}{q} \]
- From \(C_1\) and \(C_2\):
\[ \frac{-1}{-1} = 1 \]

5. **Equate the ratios**:
Since the angle bisector condition holds, we can equate:
\[ \frac{p}{q} = 1 \implies p = q \]

6. **Conclusion**:
Thus, the condition for the lines to bisect each other's angles is:
\[ p = q \]