If (a^2 + b^2) / c^2 = (b^2 + c^2) / a^2 = (c^2 + a^2) / b^2 = k, where k ≠ 0 and a, b, c are non-zero real numbers, then find the exact value of k.

Introduction / Context: This question is a classic symmetric-algebra simplification problem. You are told that three different expressions are equal to the same constant k. Each expression compares a sum of two squares to the remaining square, but in a cyclic way. Instead of trying to solve for a, b, and c individually, the smart approach is to rewrite each equality in a comparable form and then add them. The symmetry makes many terms combine neatly, allowing k to be determined uniquely under the non-zero assumption.

Given Data / Assumptions:

• (a^2 + b^2) / c^2 = k • (b^2 + c^2) / a^2 = k • (c^2 + a^2) / b^2 = k • k ≠ 0, and a, b, c are non-zero real numbers • Required: value of k

Concept / Approach: Convert each ratio equation into a linear relation in squares by multiplying both sides by the denominator. Then add the three resulting equations. The left side becomes 2(a^2 + b^2 + c^2), while the right side becomes k(a^2 + b^2 + c^2). Since a, b, c are non-zero, the sum a^2 + b^2 + c^2 is strictly positive, so we can divide by it safely to get k.

Step-by-Step Solution: 1) From (a^2 + b^2)/c^2 = k, multiply both sides by c^2: a^2 + b^2 = k * c^2 2) From (b^2 + c^2)/a^2 = k, multiply both sides by a^2: b^2 + c^2 = k * a^2 3) From (c^2 + a^2)/b^2 = k, multiply both sides by b^2: c^2 + a^2 = k * b^2 4) Add all three equations (left sides together, right sides together): (a^2 + b^2) + (b^2 + c^2) + (c^2 + a^2) = k(c^2 + a^2 + b^2) 5) Combine like terms on the left: 2a^2 + 2b^2 + 2c^2 = k(a^2 + b^2 + c^2) 6) Factor out 2 on the left: 2(a^2 + b^2 + c^2) = k(a^2 + b^2 + c^2) 7) Since a, b, c are non-zero, a^2 + b^2 + c^2 > 0, so divide both sides by it: k = 2

Verification / Alternative check: Pick any convenient non-zero values with symmetry, for example a = b = c = 1: (a^2 + b^2)/c^2 = (1 + 1)/1 = 2, and similarly the other two ratios are also 2. This confirms that k = 2 is consistent and matches the derived unique value.

Why Other Options Are Wrong: • 1 or 1/2: would contradict the summed relation 2(a^2 + b^2 + c^2) = k(a^2 + b^2 + c^2). • 3 or 4: would require the left side to be 3 or 4 times the sum of squares, which is impossible because the left side is exactly 2 times that sum.

Common Pitfalls: • Trying to solve for a, b, c individually instead of using symmetry. • Forgetting that a^2 + b^2 + c^2 is positive when a, b, c are real and non-zero.

Final Answer: 2