To find the value of pq, we need to determine the angle bisectors of the given pairs of straight lines.

Given pair 1: x^2 - 2pxy - y^2 = 0
Given pair 2: x^2 - 2qxy - y^2 = 0

Let's analyze each pair separately to find their angle bisectors.

Angle Bisectors for Pair 1:
The equation of a line passing through the origin at an angle θ with the x-axis is given by:
tan(θ) = 2p / (1 - p^2)

To find the angle bisector of the given pair 1, we need to find the angle (θ1) for which this equation holds true.

Angle Bisectors for Pair 2:
Similarly, the equation of a line passing through the origin at an angle φ with the x-axis is given by:
tan(φ) = 2q / (1 - q^2)

To find the angle bisector of the given pair 2, we need to find the angle (φ1) for which this equation holds true.

Angle Bisectors for Pair 1 bisecting Pair 2:
Now, let's find the angle bisector of pair 1 that bisects the angle between pair 2. Let this angle be α.

The equation of the angle bisector of pair 1 that bisects the angle between pair 2 is given by:
tan(α) = (tan(θ1) + tan(φ1)) / (1 - tan(θ1) * tan(φ1))

Since this line bisects the angle between pair 1 and pair 2, it should also be the angle bisector for pair 2 bisecting pair 1.

Hence, we can equate α with the angle bisector of pair 2, which is φ1.

Solving the equation tan(α) = (tan(θ1) + tan(φ1)) / (1 - tan(θ1) * tan(φ1)) = tan(φ1), we can find the value of α.

Now, the angle bisector of pair 1 bisecting pair 2 is α, and the angle bisector of pair 2 bisecting pair 1 is φ1.

Since these angles are bisectors, they should be equal.

Therefore, α = φ1

Simplifying the equation, we get:
(tan(θ1) + tan(φ1)) / (1 - tan(θ1) * tan(φ1)) = tan(φ1)

Cross-multiplying and simplifying, we have:
tan(θ1) + tan(φ1) = tan(φ1) - tan(θ1) * tan(φ1)^2

Rearranging the terms, we get:
2 * tan(φ1) * tan(θ1) - tan(θ1) - tan(φ1) = 0

This equation should hold true for any value of θ1 and φ1.

Comparing the coefficients, we get:
2 * tan(φ1) * tan(θ1) = 1
tan(θ1) = 1 / (2 * tan(φ1))

Comparing this equation with the equation of the angle bisector for pair 1, we have: