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

By shifting the origin to the point $(2, 3)$ and then rotating the coordinate axes through an angle $\theta$ in the counter-clockwise direction,if the equation $3x^2 + 2xy + 3y^2 - 18x - 22y + 50 = 0$ is transformed to $4X^2 + 2Y^2 - 1 = 0$,then the angle $\theta =$

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:

If the equation of a curve $C$ is transformed to $9x^2 + 25y^2 = 225$ by the rotation of the coordinate axes about the origin through an angle $\frac{\pi}{4}$ in the positive direction,then the equation of the curve $C$ before the transformation 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.

By shifting the origin to the point $(-1, 2)$ through translation of axes,if $ax^2+2hxy+by^2+2gx+2fy+c=0$ is the transformed equation of $2x^2-xy+y^2-3x+4y-5=0$,then $2(f+g+h)=$