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.
Aptitude
Elementary Algebra
Difficulty: Easy
Choose an option
-
A10
-
B2
-
C4
-
D5
Answer
Correct Answer: 10
Explanation
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:
Compute LHS: (ax + by)^2 + (bx − ay)^2 = 6^2 + 2^2 = 36 + 4 = 40.By identity, LHS = (a^2 + b^2)(x^2 + y^2) = (a^2 + b^2) * 4.Thus (a^2 + b^2) * 4 = 40 ⇒ a^2 + b^2 = 10.Verification / Alternative check:
Choose any (x, y) with x^2 + y^2 = 4 and construct a, b to match the given linear forms; the identity ensures the same result.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:
10