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If x = a cosθ + b sinθ and y = b cosθ – a sinθ, then x² + y² is equal to
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- ab
- a² + b²
- a² – b²
- 1
Correct Option: B
x = a cosθ + b sinθ .... (i)
y = b cosθ – a sinθ ... (ii)
On squaring both equations and adding.
x² + y² = a² cos² θ + b² sin²θ + 2ab sin θ . cos θ + b² cos² θ + a² sin² θ - 2ab sin θ . cosθ
2ab sinθ . cosθ + b² cos² + a² sin²θ - 2ab sinθ . cosθ
= a² cos²θ + a² sin²θ + b² sin²θ + b² cos²θ
= a² (cos²θ + sin²θ) + b² (sin²θ + cos²θ)
= a² + b² [∵ cos²θ + sin²θ]
