Using orthogonal-sum pattern: Given ax + by = 6, bx − ay = 2, and x^2 + y^2 = 4, find the value of a^2 + b^2.

Introduction / Context:This problem uses a standard identity: (ax + by)^2 + (bx − ay)^2 = (a^2 + b^2)(x^2 + y^2). Recognizing this pattern allows immediate evaluation of a^2 + b^2 without solving for x or y individually.

Given Data / Assumptions:

• ax + by = 6
• bx − ay = 2
• x^2 + y^2 = 4
• Find a^2 + b^2

Concept / Approach:The expression (ax + by, bx − ay) behaves like the image of (x, y) under a scaled rotation. Squaring and adding corresponds to preserving x^2 + y^2 up to the scale factor a^2 + b^2. Hence compute LHS and divide by x^2 + y^2 to get the desired sum of squares of coefficients.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 2, 4, 5: Do not satisfy the proportionality with the computed 40 on the left and 4 on the right.

Common Pitfalls:

• Forgetting that cross terms cancel when adding the two squares.
• Arithmetic error in 6^2 + 2^2.

Final Answer: