If the axes are rotated by an angle of $30^{\circ}$ in the counter-clockwise direction,find the coordinates of the point $(4, -2\sqrt{3})$ with respect to the new axes.

The transformed equation of $3x^2 - 6xy + 8y^2 = 8$ when the axes are rotated about the origin through an angle $\frac{\pi}{4}$ in the positive direction is:

Let $L$ be the line $2x + y = 2$. If the axes are rotated by $45^\circ$,then the intercepts made by the line $L$ on the new axes are respectively

When the coordinate axes are rotated through an angle $\frac{\pi}{4}$ in the positive direction,an equation is transformed to $x^2+y^2-6x+8y+21=0$. Then the original equation is

If the point $P(1,3)$ undergoes the following transformations successively:
$(i)$ Reflection with respect to the line $y=x$.
(ii) Translation through $3$ units along the positive direction of the $X$-axis.
(iii) Rotation through an angle of $\frac{\pi}{6}$ about the origin in the clockwise direction.
Then,the final position of the point $P$ is

If the origin is shifted to $(2,3)$ and the axes are rotated through an angle $45^{\circ}$ about that point,then the transformed equation of $2 x^2+2 y^2-8 x-12 y+18=0$ is