2021 · Q25
If x/(b + c − a) = y/(b − c − a) = z/(a + b − c) = k, then what is x² + y² + z² − 2xy − 2yz + 2zx equal to ?
Elementary MathematicsRatio and Proportion
- Ak²(a² + b² + c²)
- Bk²(a² − b² + c²)
- Ck²(a + b + c)²
- Dk²(a − b + c)²
Explanation
Write x = k(b+c−a), y = k(b−c−a), z = k(a+b−c). The expression x² + y² + z² − 2xy − 2yz + 2zx is exactly (x − y + z)², as seen by expanding. Now x − y + z = k[(b+c−a) − (b−c−a) + (a+b−c)] = k(a + b + c). Squaring gives k²(a + b + c)², option (c).