Let S : sqrt3x^2-4xy+sqrt3y^2=0.
:. sqrt3x^2-3xy-xy+sqrt3y^2=0.
:. sqrt3x(x-sqrt3y)-y(x-sqrt3y)=0.
:. (sqrt3x-y)(x-sqrt3y)=0.
Thus, the individual lines l_1 and l_2" of "S are,
l_1 : sqrt3x-y=0, and, l_2 : x-sqrt3y=0.
Observe that, O(0,0) in l_1 : y=sqrt3x=xtan(pi/3), and, is
making an /_" of "pi/3 with the +ve direction of the X- Axis.
So, if S is rotated anti-clockwise by an /_" of "pi/6 about O , then, so
would be the effect on both l_1 and l_2.
This means that, after the said rotation, l_1 will now make an
/_" of "(pi/3+pi/6)=pi/2 with the +ve direction of the X- Axis.
In other words, it will become the Y-Axis, or, x=0.
Similarly, l_2 becomes, y=xtan(pi/6+pi/6)=sqrt3x.
Hence, the combined eqn. of lines becomes
x(sqrt3x-y)=0, i.e., sqrt3x^2-xy=0.
Enjoy Maths.!