When the equation $x^2 - 3xy + 11x - 12y + 36 = 0$ is transformed by shifting the origin to the point $(-4, 1)$ while keeping the axes parallel,it becomes $ax^2 + bxy + 1 = 0$. Then $b^2 - a = \dots$

If the origin is shifted to the point $(1, 1)$ and the axes are rotated through an angle $45^{\circ}$ about this point,then the transformed equation of the equation $x^2 + 2xy + y^2 - 1 = 0$ is

To which point should the origin be shifted,without changing the direction of the axes,so that the equation $x^2 + y^2 - 4x + 6y - 7 = 0$ transforms into an equation that contains no first-degree terms?

If the coordinates of a point $P$ change to $(2, -6)$ when the coordinate axes are rotated through an angle of $135^{\circ}$,then the coordinates of $P$ in the original system are

$(a, b)$ is the point to which the origin has to be shifted by translation of axes so as to remove the first-degree terms from the equation $2x^2 - 3xy + 4y^2 + 5y - 6 = 0$. If the angle by which the axes are to be rotated in the positive direction about the origin to remove the $xy$-term from the equation $ax^2 + 23abxy + by^2 = 0$ is $\theta$,then $\tan 2\theta =$

The transformed equation of $3x^2+4xy+y^2-8x-4y-4=0$ is $f(X, Y)=aX^2+2hXY+bY^2+c=0$ when the origin is shifted to a new point by the translation of axes. Then $f(1,1)=$