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 $3x^2 - 4xy = r^2$ when the coordinate axes are rotated about the origin through an angle of $\operatorname{Tan}^{-1}(2)$ in the positive direction is:

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

When the coordinate axes are rotated through an angle of $45^{\circ}$ about the origin in the positive direction,if the transformed equation of a curve is $17x^2 - 16xy + 17y^2 = 225$,then the original equation of that curve is:

By rotating the coordinate axes in the positive direction about the origin by an angle $\alpha$,if the point $(1,2)$ is transformed to $\left(\frac{3 \sqrt{3}-1}{2 \sqrt{2}}, \frac{\sqrt{3}+3}{2 \sqrt{2}}\right)$ in the new coordinate system,then $\alpha=$

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 $2H(G+F)=$