Answer
This is an ellipse, with the main axis on the line $y=1$:
the center is $(1,1)$
the foci are $(2, 1)$ and $(0,1)$
the vertices are $(1\pm\sqrt{2},1)$
Work Step by Step
We group like terms
$(x^{2}-2x)+(2y^{2}-4y)=-1$
$(x^{2}-2x)+2(y^{2}-2y)=-1 \quad $
We complete the squares
$(x^{2}-2x+1)+2(y^{2}-2y+1)=-1+1+2(1) $
$(x-1)^{2}+2(y-1)^{2}=2\quad $
Divide with $2$
$\displaystyle \frac{(x-1)^{2}}{2}+(y-1)^{2}=1$
This has the form $\quad \quad \displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad (a\gt b)$
Foci on the x-axis$, a=\sqrt{2}, \quad b=1$
Center-to-focus distance: $\quad c=\sqrt{a^{2}-b^{2}}=\sqrt{2-1}=1$
Foci: $\quad(\pm 1, 0)$
Vertices: $\quad (\pm\sqrt{2}, 0)$
When $x$ is replaced with $(x-1)$, a shift to the right is made.
When $y$ is replaced with $(y-1)$, a shift upward is made.
$(x,y)\rightarrow(x+1,y+1)$
(A vertical shift $\Rightarrow$ the major axis moves up to the line $y=1$.)
The center shifts to $(0+1,0+1)=(1,1)$
The foci shift to $(1\pm 1, 1+0)\Rightarrow(2, 1)$ and $(0,1)$
The vertices shift to $(1\pm\sqrt{2},1)$
