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

When the axes are rotated through an angle $\theta$ about the origin in the anticlockwise direction and then translated to the new origin $(2, -2)$,if the transformed equation of $x^2+y^2=4$ is $X^2+Y^2+aX+bY+c=0$,then $a+b+c=$

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

If the origin is shifted to remove the first degree terms from the equation $2x^2 - 3y^2 + 4xy + 4x + 4y - 14 = 0$,then with respect to this new coordinate system,the transformed equation of $x^2 + y^2 - 3xy + 4y + 3 = 0$ is

After the coordinate axes are rotated through an angle $\frac{\pi}{4}$ in the anti-clockwise direction without shifting the origin,if the equation $x^2+y^2-2x-4y-20=0$ transforms to $ax^2+2hxy+by^2+2gx+2fy+c=0$ in the new coordinate system,then $\left|\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c\end{array}\right|=$

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