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 $\theta_1, \theta_2, \theta_3$ are respectively the angles by which the coordinate axes are to be rotated to eliminate the $xy$ term from the following equations,then the descending order of these angles is:
$A_1 = 3x^2 + 5xy + 3y^2 + 2x + 3y + 4 = 0$
$A_2 = 5x^2 + 2\sqrt{3}xy + 3y^2 + 6 = 0$
$A_3 = 4x^2 + \sqrt{3}xy + 5y^2 - 4 = 0$

Suppose the axes are to be rotated through an angle $\theta$ so as to remove the $xy$ term from the equation $3x^2+2\sqrt{3}xy+y^2=0$. Then in the new coordinate system,the equation $x^2+y^2+2xy=2$ is transformed to:

$A$ 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

If the axes are rotated through an angle $\theta = \frac{\pi}{3}$ in the clockwise direction with respect to the origin $(0, 0)$,what are the coordinates of the point $(4, 2)$ in the new system?

When the coordinate axes are rotated through an angle of $45^{\circ}$ about the origin in the positive direction,if the transformed equation of a curve is $17x^2 - 16xy + 17y^2 = 225$,then the original equation of that curve is: