If the origin is shifted to the point $(1, 1)$ and the axes are rotated through an angle $45^{\circ}$ about this point,then the transformed equation of the equation $x^2 + 2xy + y^2 - 1 = 0$ is

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 $\tan 2\theta =$

The coordinate axes are rotated about the origin in the counter-clockwise direction through an angle $60^{\circ}$. If $a$ and $b$ are the intercepts made on the new axes by a straight line whose equation referred to the original axes is $x+y=1$,then $\frac{1}{a^2}+\frac{1}{b^2}=$

The angle by which the coordinate axes are to be rotated about the origin so that the transformed equation of $\sqrt{3} x^2+(\sqrt{3}-1) x y-y^2=0$ is free from the $xy$ term is: (in $^{\circ}$)

$A$ 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:

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