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

When the coordinate axes are rotated through an angle $\theta$ in anti-clockwise direction,if the transformed equation of $x^2+y^2+2xy+2x+6y+1=0$ is $(2+\sqrt{3})X^2+2XY+(2-\sqrt{3})Y^2+aX+bY+2=0$,then $3a-b=$

The transformed equation of $x^2-y^2+2x+4y=0$ when the origin is shifted to the point $(-1, 2)$ is

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:

If the axes are rotated through an angle $45^{\circ}$ about the origin in an anticlockwise direction,then the transformed equation of $y^2=4ax$ is

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: