When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$,if the equation $25x^2+9y^2=225$ is transformed to $\alpha x^2+\beta xy+\gamma y^2=\delta$,then $(\alpha+\beta+\gamma-\sqrt{\delta})^2=$

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

If the origin is shifted to $(2,3)$ and the axes are rotated through an angle $45^{\circ}$ about that point,then the transformed equation of $2 x^2+2 y^2-8 x-12 y+18=0$ is

If $2x^2+xy-6y^2+k=0$ is the transformed equation of $2x^2+xy-6y^2-13x+9y+15=0$ when the origin is shifted to the point $(a, b)$ by translation of axes,then $k=$

When the origin is shifted to the point $(2, b)$ by translation of axes,the coordinates of the point $(a, 4)$ change to $(6, 8)$. When the origin is shifted to $(a, b)$ by translation of axes,if the transformed equation of $x^2+4xy+y^2=0$ is $X^2+2HXY+Y^2+2GX+2FY+C=0$,then $2H(G+F)=$

The point $(4, 1)$ undergoes the following three transformations successively: $(i)$ Reflection about the line $y = x$,(ii) Translation through a distance of $2$ units along the positive direction of the $x$-axis,(iii) Rotation through an angle $\pi/4$ about the origin in the anti-clockwise direction. The final position of the point is given by the coordinates: