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 $a+b+c=$

By shifting the origin to the point $(2, 3)$ and then rotating the coordinate axes through an angle $\theta$ in the counter-clockwise direction,if the equation $3x^2 + 2xy + 3y^2 - 18x - 22y + 50 = 0$ is transformed to $4X^2 + 2Y^2 - 1 = 0$,then the angle $\theta =$

The transformed equation of $3x^2 - 4xy = r^2$ when the coordinate axes are rotated about the origin through an angle of $\operatorname{Tan}^{-1}(2)$ in the positive direction is:

$A$ line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle $\theta$ keeping the origin fixed,this line $L$ has the intercepts $p$ and $q$. Then

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 angle by which axes are to be rotated without changing the origin so that the transformed equation of $x^2+4xy-y^2=0$ in new coordinates $(X, Y)$ does not contain the $XY$ term is