Question

Show that the angle of rotation of the axes to eliminate xy term in the equation ax2 + 2hxy + by2 = 0 is 1/2 Tan-1(2h/a-b) where(a≠b).

Answer 1

Let θ be the required angle of rotation.

Let (X, Y) be the new coordinates of (x, y),

So, x = X cosθ – Y sinθ, y = X sinθ + Y cosθ

=> The transformed equation is a(X cosθ – Y sinθ)2 + 2h(X cosθ – Y sinθ)(X sinθ + Y cosθ) + b(X sinθ + Y cosθ)2 =0

=> a(X2 cos2θ + Y2 sin2θ – 2XY cosθ sinθ) + 2h (X2 cosθ sinθ+ XY cos2θ – XY sin2θ – Y2 sinθ cosθ) + b(X2 sin2θ +Y2 cos2θ + 2XY sinθ cosθ) = 0

=> X2(a cos2θ + 2h cosθ sinθ + b sin2θ) + 2XY(–a cosθ sinθ + h cos2θ – h sin2θ + b sinθ cosθ) + Y2(a sin2θ – 2h sinθ cosθ + b cos2θ) = 0