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Question: Mixed term xyis to be removed from the general equation \(ax^{2} + by^{2} + 2hxy + 2gx + 2fy + c = 0...

Mixed term xyis to be removed from the general equation ax2+by2+2hxy+2gx+2fy+c=0ax^{2} + by^{2} + 2hxy + 2gx + 2fy + c = 0, one should rotate the axes through an angle θ\thetagiven by tan⁡2θ\tan 2\theta =

A

a−b2h\frac{a - b}{2h}

B

2ha+b\frac{2h}{a + b}

C

a+b2h\frac{a + b}{2h}

D

2ha−b\frac{2h}{a - b}

Answer

2ha−b\frac{2h}{a - b}

Explanation

Solution

Let (x′,y′)(x',y') be the coordinates on new axes, then put x=x′cos⁡θ−y′sin⁡θ,y=x′sin⁡θ+y′cos⁡θx = x'\cos\theta - y'\sin\theta,y = x'\sin\theta + y'\cos\thetain the equation, then the coefficient of xy in the transformed equation is 0. So, 2(b−a)2 ( b - a ) sin⁡θ.cos⁡θ+2hcos⁡2θ=0\sin\theta.\cos\theta + 2h\cos 2\theta = 0⇒ tan⁡2θ=2ha−b\tan 2\theta = \frac{2h}{a - b}