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

The transformed equation of $x^2+6xy+8y^2=10$ when the axes are rotated through an angle $\frac{\pi}{4}$ is:

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

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 transformed equation $3x^2 + 3y^2 + 2xy = 2$,when the coordinate axes are rotated through an angle $45^{\circ}$,is

After the coordinate axes are rotated through an angle $\frac{\pi}{4}$ in the anti-clockwise direction without shifting the origin,if the equation $x^2+y^2-2x-4y-20=0$ transforms to $ax^2+2hxy+by^2+2gx+2fy+c=0$ in the new coordinate system,then $\left|\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c\end{array}\right|=$