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

If $(h, k)$ is the new origin to be chosen to eliminate first degree terms from the equation $S \equiv 2x^2 - xy - y^2 - 3x + 3y = 0$ by translation and if $\theta$ is the angle with which the axes are to be rotated about the origin in anticlockwise direction to eliminate the $xy$-term from $S = 0$,then $\tan 2\theta =$

Suppose the axes are to be rotated through an angle $\theta$ so as to remove the $xy$ term from the equation $3x^2+2\sqrt{3}xy+y^2=0$. Then in the new coordinate system,the equation $x^2+y^2+2xy=2$ is transformed to:

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

Find the new coordinates of the reflection of the point $(2, -1)$ about the $y$-axis,when the axes are rotated by $45^{\circ}$ in the negative direction without shifting the origin.

If the origin is shifted to the point $(-2, 1)$,what are the new coordinates of the point $(4, -5)$?