The point to which the origin is to be shifted by translation of axes so that the transformed equation of $y^2+4y+8x-2=0$ will not contain $y$ term and constant term is

If the axes are rotated through an angle $\alpha$,then the number of values of $\alpha$ such that the transformed equation of $x^2+y^2+2x+2y-5=0$ contains no linear terms is

If the axes are rotated through an angle $45^{\circ}$ in the positive direction without changing the origin,then the coordinates of the point $(\sqrt{2}, 4)$ in the old system are

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

If the area of the region bounded by the curves $y=x^2$ and $x=y^2$ is $k$,then the area of the region bounded by the curves $\frac{x+\sqrt{3} y}{2}=\left(\frac{\sqrt{3} x-y}{2}\right)^2$ and $\frac{\sqrt{3} x-y}{2}=\left(\frac{x+\sqrt{3} y}{2}\right)^2$ 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