When the origin is shifted to $(h, k)$ by translation of axes,the transformed equation of $x^2+2x+2y-7=0$ does not contain the $x$ term and the constant term. Then $(2h+k) =$

Find the transformed equation of $x \cos \theta + y \sin \theta = p$,when the axes are rotated through an angle $\theta$.

The point $P(4,1)$ undergoes the following transformations in succession:
$(i)$ origin is shifted to the point $(1,6)$ by translation of axes
(ii) translation through a distance of $2$ units along the positive direction of $X$-axis
(iii) rotation of axes through an angle of $90^{\circ}$ in the positive direction
Then the coordinates of the point $P$ in its final position are

If the axes are rotated through an angle $45^{\circ}$,then the coordinates of the point $(4 \sqrt{2}, -6 \sqrt{2})$ in the new system are . . . . . .

The transformed equation of $3x^2 - 4xy = r^2$ when the coordinate axes are rotated about the origin through an angle of $\operatorname{Tan}^{-1}(2)$ in the positive direction is:

By rotating the axes about the origin in an anti-clockwise direction by a certain angle,if the equation $x^2+4xy+y^2=1$ is transformed to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,then find the value of $\sqrt{\frac{a^2+b^2}{a^2}}$.