The transformed equation of $x^2+y^2=r^2$ when the axes are rotated through an angle $36^{\circ}$ is

The point to which the origin should be shifted so that the equation $y^2-6y-4x+13=0$ will not contain any term in $y$ and the constant term,is

Without changing the direction of the axes, the origin is shifted to the point $(2, 3)$. Then the equation $x^{2} + y^{2} - 4x - 6y + 9 = 0$ changes to

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 $\left|\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c\end{array}\right|=$

If the origin is shifted to $(1, -2)$ and the axes are rotated by an angle of $30^{\circ}$,what will be the new coordinates of $(1, 1)$?

If $(a, b)$ are the new coordinates of the point $(2, 3)$ after shifting the origin to the point $(3, 2)$ by translation of axes,and $(c, d)$ are the new coordinates of the point $(a, b)$ after rotating the axes through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction,then find the value of $d-c$.