When the origin is shifted to $(-1, 2)$ by the translation of axes,the transformed equation of $x^2+y^2+2x-4y+1=0$ is

The transformed equation of $x^2-y^2+2x+4y=0$ when the origin is shifted to the point $(-1, 2)$ is

When the origin is shifted to the point $(2, b)$ by translation of axes,the coordinates of the point $(a, 4)$ change to $(6, 8)$. When the origin is shifted to $(a, b)$ by translation of axes,if the transformed equation of $x^2+4xy+y^2=0$ is $X^2+2HXY+Y^2+2GX+2FY+C=0$,then $2H(G+F)=$

If the axes are rotated through an angle $45^{\circ}$,the coordinates of the point $(2 \sqrt{2}, -3 \sqrt{2})$ in the new system are

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.

The transformed equation of $3x^2 - 4xy = r^2$ when the coordinate axes are rotated through an angle $\tan^{-1}(2)$ is: