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

Suppose the new axes $X, Y$ are generated by rotating the coordinate axes $x, y$ about the origin through an angle of $30^{\circ}$ in the anti-clockwise direction. Then,the transformed equation of $x^2+2 \sqrt{3} xy - y^2 = 2a^2$ with respect to the new axes $X, Y$ is

When the origin is shifted to the point $(2,3)$ and then the coordinate axes are rotated through an angle $\frac{\pi}{3}$ in the counter-clockwise sense,then the transformed equation of $3 x^2+2 x y+3 y^2-18 x-22 y+50=0$ is

If the coordinate axes are rotated in the positive direction by $45^{\circ}$ without changing the origin,then the transformed equation of $3x^2 + 3y^2 + 2xy - 2 = 0$ is

The transformed equation of $3x^2 + 3y^2 + 2xy = 2$,when the coordinate axes are rotated through an angle of $45^{\circ}$,is

The transformed equation of $x^2+6xy+8y^2=10$ when the axes are rotated through an angle $\frac{\pi}{4}$ is: