The transformed equation of 3 x 2 - 4 x y = r 2 , when the coordinate axes are rotated through an angle tan - 1 2 is

Question

The transformed equation of 3 x 2 - 4 x y = r 2 , when the coordinate axes are rotated through an angle tan - 1 2 is, X 2 - 4 Y 2 = r 2, 2 X Y + r 2 = 0, 4 Y 2 - X 2 = r 2, X Y = r 2

MARKS App · Accepted Answer

θ = tan - 1 2 ⇒ tan θ = 2 cos θ = 1 5 , sin θ = 2 5 x = X cos θ - Y sin θ = X - 2 Y 5 , y = X sin θ + Y cos θ = 2 X + Y 5 ⇒ 4 x y - 3 x 2 = r 2 ⇒ 4 X - 2 Y 5 2 X + Y 5 - 3 X - 2 Y 5 2 = r 2 ⇒ X 2 - 4 Y 2 = r 2

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