To solve this problem, let's break it down into steps:

Step 1: Find the equations of the lines xy = 0 and (1 - m^2)xy - mx^2 = 0
Step 2: Find the angle between the lines xy = 0 and (1 - m^2)xy - mx^2 = 0
Step 3: Determine if one of the lines in (1 - m^2)xy - mx^2 = 0 is a bisector of the angle between the lines xy = 0
Step 4: Determine the value of m that satisfies the conditions of the problem

Step 1: Find the equations of the lines xy = 0 and (1 - m^2)xy - mx^2 = 0
The equation xy = 0 represents two lines: x = 0 and y = 0
The equation (1 - m^2)xy - mx^2 = 0 can be rearranged as (1 - m^2)xy = mx^2
If xy ≠ 0, then (1 - m^2) = m or 1 - m^2 = 0
If (1 - m^2) = m, then m^2 + m - 1 = 0
If 1 - m^2 = 0, then m = ±1

Step 2: Find the angle between the lines xy = 0 and (1 - m^2)xy - mx^2 = 0
The angle between two lines can be found using the formula tanθ = |(m1 - m2) / (1 + m1m2)|
For the lines xy = 0, the slope is undefined (vertical line), so m1 = ∞
For the lines (1 - m^2)xy - mx^2 = 0, the slope is given by m2 = (1 - m^2) / m
Substituting these values into the formula, we get tanθ = |(∞ - (1 - m^2) / m) / (1 + (∞)(1 - m^2) / m)|
Simplifying, we have tanθ = |m^2 - 1|

Step 3: Determine if one of the lines in (1 - m^2)xy - mx^2 = 0 is a bisector of the angle between the lines xy = 0
We know that the angle between the lines xy = 0 is 90 degrees. Therefore, if one of the lines in (1 - m^2)xy - mx^2 = 0 is a bisector, then the angle between the lines is 45 degrees.
Since the tangent of 45 degrees is 1, we can equate |m^2 - 1| to 1 and solve for m:
m^2 - 1 = 1 or m^2 - 1 = -1
Solving these equations, we get m = ±√2

Step 4: Determine the value of m that satisfies the conditions of the problem
From step 1, we found that m can be ±1 or ±√2. However, since the problem states that m > 0, we can eliminate m = -1 and m = -√2.
Therefore,