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If x: a = y: b = z: c, is the identity (a² + b² + c²)(x² + y² + z²) = (ax + by + cz)² true?

1. True
2. False
3. Only if a = b = c
4. Only if x = y = z

Correct answer: True

Solution

Let x = ka, y = kb, z = kc. LHS = (a² + b² + c²)(k² a² + k² b² + k² c²) = k² (a² + b² + c²)². RHS = (a*ka + b*kb + c*kc)² = (k(a² + b² + c²))² = k² (a² + b² + c²)². LHS = RHS, so the identity is true (it is the equality case of Cauchy-Schwarz).

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