If the origin of a coordinate system is shifted to $(-\sqrt{2}, \sqrt{2})$ and the coordinate system is rotated anti-clockwise through an angle $45^{\circ}$,then the point $P(1, -1)$ in the original system has new coordinates

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:
$A_1 = 3x^2 + 5xy + 3y^2 + 2x + 3y + 4 = 0$
$A_2 = 5x^2 + 2\sqrt{3}xy + 3y^2 + 6 = 0$
$A_3 = 4x^2 + \sqrt{3}xy + 5y^2 - 4 = 0$

When the origin is shifted to $(h, k)$ by translation of axes,the transformed equation of $x^2+2x+2y-7=0$ does not contain the $x$ term and the constant term. Then $(2h+k) =$

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

When the origin is shifted to the point $P$ by translation of axes,the equation $2x^2+y^2-4x+4y=0$ is transformed to $2x^2+y^2-8x+8y+18=0$. Then the transformed equation of the straight line $x+2y+2=0$ if the origin is shifted to the same point $P$ is

If the transformed equation of the equation $2x^2+3xy-2y^2-17x+6y+8=0$ after translating the coordinate axes to a new origin $(\alpha, \beta)$ is $aX^2+2hXY+bY^2+c=0$,then $3\alpha+c=$