If the transformed equation of the equation $2x^2+3xy-2y^2-17x+6y+8=0$ after translating the coordinate axes to a new origin $(\alpha, \beta)$ is $aX^2+2hXY+bY^2+c=0$,then $3\alpha+c=$

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?

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$,if the equation $49x^2+25y^2=1225$ is transformed to $px^2+qxy+ry^2=t$ and the $G.C.D$ of $p, q, r, t$ is $1$,then:

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

When the origin is shifted to $(1, -2)$ by translation of coordinate axes,the transformed coordinates of $(3, -2)$ are $(\alpha, \beta)$. If the axes are rotated about the origin through an angle of $45^{\circ}$ after the translation,then the transformed coordinates of $(\alpha, \beta)$ are

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 $d-c$.