Question

The angle of rotation of axes to remove xy term in the equation 3x^2 – 2√3xy +9y^2 = 10 is​

Answer 1

Given: the equation $3x^{2}-2\sqrt{3}xy+9y^{2}=10$

To find: the angle of rotation of axes to remove the $xy$ term

Solution:

Let the angle of rotation be $\theta$

The the rotational formulae be:

• $x=X\:cos\theta-Y\:sin\theta$
• $y=X\:sin\theta+Y\:cos\theta$

Substituting them in the given equation, we get

$\quad 3(X\:cos\theta-Y\:sin\theta)^{2}-2\sqrt{3}(X\:cos\theta-Y\:sin\theta)(X\:sin\theta+Y\:cos\theta)+9(X\:sin\theta+Y\:cos\theta)^{2}=10$

In order to remove the $xy$ equivalent $XY$ terms, we must have:

$\quad -6\:sin\theta\:cos\theta-2\sqrt{3}\:(cos^{2}\theta-sin^{2}\theta)+18\:sin\theta\:cos\theta=10$

$\Rightarrow 12\:sin\theta\:cos\theta=2\sqrt{3}\:(cos^{2}\theta-sin^{2}\theta)$

$\Rightarrow 6\:sin2\theta=2\sqrt{3}\:cos2\theta$

$\Rightarrow tan2\theta=\sqrt{3}$

$\Rightarrow tan2\theta=tan60^{\circ}$

$\Rightarrow 2\theta=60^{\circ}$

$\Rightarrow \theta=30^{\circ}$

Answer: angle of rotation is $30^{\circ}$