Answer
$x^2+\frac{(y-1)^2}{2}=1$
Ellipse: center $(0, 1)$, foci $F_1(0,0)$ and $F_2(0,2)$, vertices $(0, 1\pm\sqrt 2)$
lengths of the major axis $2\sqrt 2$, length of the minor axis $22$
See graph.
Work Step by Step
Step 1. Form square with variables: $2x^2+y^2-2y+1=2$ or $2x^2+(y-1)^2=2$. Divide both sides by 2 to get $x^2+\frac{(y-1)^2}{2}=1$
Step 2. We can identify the above equation as an ellipse with a center at $(0, 1)$
Step 3. The major axis is along the y-axis, $a=\sqrt 2, b=1$ we have $c=\sqrt {2-1}=1$, so the foci are $(0,1\pm1)$ or $F_1(0,0)$ and $F_2(0,2)$
Step 4. The vertices are $(0, 1\pm\sqrt 2)$
Step 5. The lengths of the major axis is $2a=2\sqrt 2$ and the length of the minor axis is $2b=2$
Step 6. See graph.
