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 $f(1,1)=$

$(a, b)$ 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 $\tan 2\theta =$

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

Suppose the new axes $X, Y$ are generated by rotating the coordinate axes $x, y$ about the origin through an angle of $30^{\circ}$ in the anti-clockwise direction. Then,the transformed equation of $x^2+2 \sqrt{3} xy - y^2 = 2a^2$ with respect to the new axes $X, Y$ is

Find the coordinates of $M$ in the original system if the point $M$ changes to $(4, -3)$ when the axes are rotated through an angle of $135^{\circ}$.

If $(h, k)$ is the new origin to be chosen to eliminate first degree terms from the equation $S \equiv 2x^2 - xy - y^2 - 3x + 3y = 0$ by translation and if $\theta$ is the angle with which the axes are to be rotated about the origin in anticlockwise direction to eliminate the $xy$-term from $S = 0$,then $\tan 2\theta =$