The point to which the origin should be shifted so that the equation $y^2-6y-4x+13=0$ will not contain any term in $y$ and the constant term,is

When the coordinate axes are rotated through an angle $\theta$ in anti-clockwise direction,if the transformed equation of $x^2+y^2+2xy+2x+6y+1=0$ is $(2+\sqrt{3})X^2+2XY+(2-\sqrt{3})Y^2+aX+bY+2=0$,then $3a-b=$

If the axes are rotated through an angle $45^{\circ}$ in the positive direction without changing the origin,then the coordinates of the point $(\sqrt{2}, 4)$ in the old system are

When the origin is shifted to the point $\left(\frac{3}{2}, \frac{3}{2}\right)$ by the translation of coordinate axes,then the transformed equation of $32 x^2+8 x y+32 y^2-108 x-108 y+99=0$ is

The point $P(4,1)$ undergoes the following transformations in succession:
$(i)$ origin is shifted to the point $(1,6)$ by translation of axes
(ii) translation through a distance of $2$ units along the positive direction of $X$-axis
(iii) rotation of axes through an angle of $90^{\circ}$ in the positive direction
Then the coordinates of the point $P$ in its final position are

If the axes are rotated by an angle of $30^{\circ}$ in the negative direction (clockwise) while keeping the origin fixed,what are the new coordinates of the point $(2, 1)$?