If x = a cos θ + b sin θ and y = a sin θ – b cos θ

If x = a cos θ + b sin θ and y = a sin θ – b cos θ, then what is x2 + y2 equal to?

  1. 2ab
  2. a + b
  3. a2 + b2
  4. a2 – b2

Answer

x2 + y2 = (a cos θ + b sin θ)2 + (a sin θ – b cos θ)2

cos2θ + sin2θ = 1

x2 + y2 = a2cos2θ + b2sin2θ + 2ab cos θ sin θ + a2sin2θ + b2cos2θ - 2ab cos θ sin θ

x2 + y2 = a2(cos2θ + sin2θ) + b2(cos2θ + sin2θ)

x2 + y2 = a2 + b2

The correct option is C.