If ax + by = 1 and bx + ay = 2ab / (a^2 + b^2) for real numbers a and b with a^2 + b^2 ≠ 0, then what is the value of (x^2 + y^2)(a^2 + b^2)?

Introduction / Context:
This question involves solving a simple linear system in two variables x and y and then evaluating a symmetric quadratic expression (x^2 + y^2)(a^2 + b^2). The parameters a and b are arbitrary real numbers, so the goal is to find a result that is independent of their specific values. This tests both linear algebra skills and algebraic manipulation.

Given Data / Assumptions:

• ax + by = 1.
• bx + ay = 2ab / (a^2 + b^2).
• a and b are real numbers with a^2 + b^2 ≠ 0.
• We need the value of (x^2 + y^2)(a^2 + b^2).

Concept / Approach:
We first solve the linear system for x and y using standard methods such as elimination or solving as a 2×2 system. Once explicit expressions for x and y are found, we compute x^2 + y^2 and multiply by a^2 + b^2. The algebra simplifies beautifully to a constant value independent of a and b.

Step-by-Step Solution:
Write the system: (1) ax + by = 1. (2) bx + ay = 2ab / (a^2 + b^2). Solve for x and y. Treat this as a matrix equation: [a b][x] = [1] [b a][y] [2ab/(a^2 + b^2)]. Compute the determinant: Δ = a^2 − b^2. Using Cramer's rule, x = [1·a − b·(2ab/(a^2 + b^2))] / Δ, and y = [a·(2ab/(a^2 + b^2)) − b·1] / Δ. Simplify carefully to obtain: x = a / (a^2 + b^2), and y = b / (a^2 + b^2). Now compute x^2 + y^2: x^2 + y^2 = a^2/(a^2 + b^2)^2 + b^2/(a^2 + b^2)^2. So x^2 + y^2 = (a^2 + b^2)/(a^2 + b^2)^2 = 1/(a^2 + b^2). Finally, (x^2 + y^2)(a^2 + b^2) = [1/(a^2 + b^2)]·(a^2 + b^2) = 1.
Verification / Alternative check:
Choose simple values such as a = 1, b = 1. Then the equations become x + y = 1 and x + y = 1, and the second equation gives 2·1·1/(1 + 1) = 1, which is consistent. Any pair (x, y) with x + y = 1 works, but the derived specific solution gives x = 1/2, y = 1/2. Then x^2 + y^2 = 1/4 + 1/4 = 1/2, and (x^2 + y^2)(a^2 + b^2) = (1/2)·2 = 1, confirming the result.

Why Other Options Are Wrong:
Values like 2 or 0.5 would mean (x^2 + y^2)(a^2 + b^2) changes with different choices of a and b, which contradicts the algebraic result. Zero would require x and y to be zero, which is impossible since ax + by = 1.

Common Pitfalls:
Mistakes usually arise from incorrect elimination or mismanaging denominators during simplification. Another frequent problem is to incorrectly interpret the second equation and omit the denominator a^2 + b^2 when rewriting it. Careful algebra and verification with sample values help avoid such errors.

Final Answer:
The value of (x^2 + y^2)(a^2 + b^2) is 1.