The transformed equation $3x^2 + 3y^2 + 2xy = 2$,when the coordinate axes are rotated through an angle $45^{\circ}$,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

The point to which the origin is to be shifted by translation of axes so that the transformed equation of $y^2+4y+8x-2=0$ will not contain $y$ term and constant term 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 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 coordinates of the point $(3,-7,5)$ in the new system,when the origin is shifted to the point $(-1,-1,-1)$ by the translation of axes,are