Question

The angle of rotation of axes in order to eliminate $xy$ term in the equation ${{x}^{2}}+2\sqrt{3}xy-{{y}^{2}}=2{{a}^{2}}$is
A.$\dfrac{\pi }{6}$
B.$\dfrac{\pi }{4}$
C.$\dfrac{\pi }{3}$
D.$\dfrac{\pi }{2}$

Answer 1

Here we have to calculate the angle of rotation of axes. For that, we will first the angle of rotation of axes to be $\phi$ and new coordinates be $\left( x',y' \right)$ . We will find the value of old coordinates in terms of new coordinates and $\phi$. We will put the value of old coordinates in the given equation and then we will equate the coefficient of $x'y'$ with zero. From there, we will get the value of angle of rotation of axes.

Complete step-by-step answer:
Let the angle of rotation of axes be $\phi$ and let the new coordinates be $x'$ and $y'$.
Thus, the old coordinates are:-
$x=x'\cos \phi -y'\sin \phi$
$y=x'\sin \phi +y'\cos \phi$
Now, we will put the value of coordinates in the given equation ${{x}^{2}}+2\sqrt{3}xy-{{y}^{2}}=2{{a}^{2}}$
${{\left( x'\cos \phi -y'\sin \phi \right)}^{2}}+2\sqrt{3}\left( x'\cos \phi -y'\sin \phi \right)\left( x'\sin \phi +y'\cos \phi \right)-{{\left( x'\sin \phi +y'\cos \phi \right)}^{2}}=2{{a}^{2}}$
On further simplification, we get the coefficient of $x'y'$ as
$\Rightarrow 2\sqrt{3}\left( {{\cos }^{2}}\phi -{{\sin }^{2}}\phi \right)-4\cos \phi \sin \phi$
Now, we will equate this coefficient with zero to eliminate $x'y'$ from the equation.
$\Rightarrow 2\sqrt{3}\left( {{\cos }^{2}}\phi -{{\sin }^{2}}\phi \right)-4\cos \phi \sin \phi =0........\left( 1 \right)$
We know from the trigonometric identities that
$\Rightarrow {{\cos }^{2}}\phi -{{\sin }^{2}}\phi =\cos 2\phi$ and$2\cos \phi \sin \phi =\sin 2\phi$
We will put these values in equation 1.
$\Rightarrow 2\sqrt{3}\cos 2\phi -2\sin 2\phi =0$
On further simplifying the terms, we get
$\Rightarrow \tan 2\phi =\sqrt{3}$
Therefore, the value of $\phi$is:-
$\Rightarrow$ $2\phi =\dfrac{\pi }{3}$
$\Rightarrow$ $\phi =\dfrac{\pi }{6}$

Hence, the required angle of rotation of axes is $\dfrac{\pi }{6}$.
Thus, the correct option is A.

Note: We had to eliminate the term from the equation ${{x}^{2}}+2\sqrt{3}xy-{{y}^{2}}=2{{a}^{2}}$ to calculate the angle of rotation of axes. So to eliminate any terms from the equation means to equate the coefficient of that term with zero.
Rotation means to turn the figure about a fixed point and that fixed point is called a center of rotation.
The shape and size of the figure remains the same after rotation but its direction gets changed. Rotation can be clockwise or anticlockwise.