(Solved): The Cartesian coordinate system Oxy is shifted along the vector (01) and rotated about the origi ...

The Cartesian coordinate system $Oxy$ is shifted along the vector $\left(\begin{array}{l}0\\ 1\end{array}\right)$ and rotated about the origin through an angle $?$ to give a new coordinate system $O^{?}{x}^{?}{y}^{?}$. In the $Oxy$ coordinate system, a line $?$ in ${\mathbb{R}}^2$ has equation $x+y?5+\sqrt{2}=0.$ (i) Find the angle $?$ such that the transformed equation of the line $?$ in the new coordinate system $O^{?}{x}^{?}{y}^{?}$ has equation $y^{?}=k$, for some constant $k$. Hence write down the transformed equation of $?$. $\{8$ marks $\}$ (ii) A curve $\mathcal{C}$ has equation $x^2+y^2?2xy+\sqrt{2}x+\sqrt{2}y?1?\sqrt{2}=0$ in the $Oxy$ coordinate system. Find the transformed equation of $\mathcal{C}$ in the $O^{?}{x}^{?}{y}^{?}$ coordinate system obtained above. $\{8$ marks $\}$ (iii) Hence, deduce that that the line $?$ and the curve $\mathcal{C}$ do not intersect. $\{3$ marks $\}$