The transformed equation of $x^2+6xy+8y^2=10$ when the axes are rotated through an angle $\frac{\pi}{4}$ is:

$A$ vector $\vec{a} = 2\hat{i} + 3\hat{j} + 7\hat{k}$ is given in a right-handed rectangular coordinate system. If the coordinate system is rotated about the $z-$axis from the positive $x-$axis to the positive $y-$axis through an angle of $\pi / 2$,then the new components of $\vec{a}$ will be:

When the coordinate axes are rotated about the origin through an angle $\frac{\pi}{4}$ in the positive direction,the equation $ax^2+2hxy+by^2=c$ is transformed to $25x^2+9y^2=225$,then $(a+2h+b-\sqrt{c})^2=$

The transformed equation of $3x^2 - 6xy + 8y^2 = 8$ when the axes are rotated about the origin through an angle $\frac{\pi}{4}$ in the positive direction is:

The point to which the origin is to be shifted by translation of axes so that the transformed equation of $y^2+4y+8x-2=0$ will not contain $y$ term and constant term is

When the coordinate axes are rotated through an angle $\theta$ in anti-clockwise direction,if the transformed equation of $x^2+y^2+2xy+2x+6y+1=0$ is $(2+\sqrt{3})X^2+2XY+(2-\sqrt{3})Y^2+aX+bY+2=0$,then $3a-b=$