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 $a+b+c=$

If $(h, k)$ is the new origin to be chosen to eliminate first degree terms from the equation $S \equiv 2x^2 - xy - y^2 - 3x + 3y = 0$ by translation and if $\theta$ is the angle with which the axes are to be rotated about the origin in anticlockwise direction to eliminate the $xy$-term from $S = 0$,then $\tan 2\theta =$

Find the transformed equation of $x \cos \theta + y \sin \theta = p$,when the axes are rotated through an angle $\theta$.

If the origin is shifted to the point $\left(\frac{3}{2},-2\right)$ by the translation of axes,then the transformed equation of $2x^2+4xy+y^2+2x-2y+1=0$ is

In order to eliminate the first degree terms from the equation $4x^2+8xy+10y^2-8x-44y+14=0$,the point to which the origin has to be shifted is

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