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.!
