$A$ line makes intercepts $5$ and $7$ on the coordinate axes. The axes are rotated through an angle $\theta$ in the positive direction about the origin so that the line makes equal intercepts on the new axes,then $|\tan \theta|=$

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 $3\alpha+c=$

If $a \alpha^2+b \beta^2+c \alpha \beta+d=0$ is the transformed equation of $4 x^2+\sqrt{3} x y+5 y^2-4=0$ obtained by using $\alpha=\frac{\sqrt{3}}{2} x+\frac{y}{2}$ and $\beta=-\frac{x}{2}+\frac{\sqrt{3}}{2} y$,then $c(a+b+d)=$

$(a, b)$ 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 axes are rotated through an angle $45^{\circ}$,the coordinates of the point $(2 \sqrt{2}, -3 \sqrt{2})$ in the new system are

When the origin is shifted to the point $P$ by translation of axes,the equation $2x^2+y^2-4x+4y=0$ is transformed to $2x^2+y^2-8x+8y+18=0$. Then the transformed equation of the straight line $x+2y+2=0$ if the origin is shifted to the same point $P$ is