Question

$If$ $2cos\theta -sin\theta =xandcos\theta -3sin\theta =yprovethat2x^2+y^2-2xy=5$

Answer 1

Given (2 cosθ - sinθ) = x and (cosθ - 3 sinθ) = y
Put the values of x and y in the equation 2x2 + y2 − 2xy (LHS)
= 2(2 cosθ − sinθ)2 + (cosθ − 3 sinθ)2 − 2(2 cosθ − sinθ)(cosθ − 3 sinθ)
= 2(4cos2θ − 4cosθ sinθ + sin2θ) + (cos2θ − 6cosθ sinθ + 9sin2θ) − 2(2cos2θ − 7cosθ sinθ + 3sin2θ)
= 8cos2θ − 8cosθ sinθ + 2sin2θ + cos2θ − 6cosθ sinθ + 9sin2θ − 4cos2θ + 14cosθ sinθ − 6sin2θ
= 5cos2θ + 5sin2θ
= 5(cos2θ + sin2θ)
= 5(1) = 5 (Since cos2θ + sin2θ = 1)
= RHS