When the axes are rotated through an angle $\theta$ about the origin in the anticlockwise direction and then translated to the new origin $(2, -2)$,if the transformed equation of $x^2+y^2=4$ is $X^2+Y^2+aX+bY+c=0$,then $a+b+c=$

If the axes are rotated by an angle of $-\pi /3$ in the negative direction and the coordinates of a point in the new system are $(4, 2)$,find the coordinates of the point in the original system.

The transformed equation of $x^2+y^2=r^2$ when the axes are rotated through an angle $36^{\circ}$ is

The transformed equation of $x^2-y^2+2x+4y=0$ when the origin is shifted to the point $(-1, 2)$ 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 coordinate axes are rotated through an angle $\frac{\pi}{6}$ about the origin,then the transformed equation of $\sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$ is