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 $\alpha=$

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

If the axes are translated to the orthocentre of the triangle formed by the points $A(7,5), B(-5,-7), C(7,-7)$,then the coordinates of the incentre of the triangle in the new system are

The point $P(1,4)$ occupies the positions $A, B$ and $C$ respectively after undergoing the following three transformations successively:
$I$. Reflection about the line $y=x$.
$II$. Translation through a distance of $1$ unit along the positive direction of $X$-axis.
$III$. Rotation of the line $OB$ through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction. Then,the coordinates of $C$ are

If the axes are rotated through an angle $\alpha$,then the number of values of $\alpha$ such that the transformed equation of $x^2+y^2+2x+2y-5=0$ contains no linear terms is

When the origin is shifted to $(2, 3)$,the transformed equation of a curve becomes $x^2+3xy-2y^2+17x-7y-11=0$. Find the original equation of the curve.