If the coordinate axes are rotated through an angle $\frac{\pi}{6}$ about the origin,then the transformed equation of $\sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$ is

If the origin is shifted to the point $(1, 1)$ and the axes are rotated through an angle $45^{\circ}$ about this point,then the transformed equation of the equation $x^2 + 2xy + y^2 - 1 = 0$ is

The new coordinates of a point $(4, 5)$,when the origin is shifted to the point $(1, -2)$ are

When the origin is shifted to $(1, -2)$ by translation of coordinate axes,the transformed coordinates of $(3, -2)$ are $(\alpha, \beta)$. If the axes are rotated about the origin through an angle of $45^{\circ}$ after the translation,then the transformed coordinates of $(\alpha, \beta)$ are

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

The origin is translated to $(1,2)$. The point $(7,5)$ in the old system undergoes the following transformations successively.
$I$. Moves to the new point under the given translation of origin.
$II$. Translated through $2$ units along the negative direction of the new $X$-axis.
$III$. Rotated through an angle $\frac{\pi}{4}$ about the origin of the new system in the clockwise direction. The final position of the point $(7,5)$ is