Evaluate the line integral over the curve of intersection

Question

Evaluate $\int_c \frac{y^2}{2}dx + zdy + xdz$, where $c$ is the curve of intersection of the plane $x+z = 1$ and ellipsoid $x^2+2y^2 + z^2 = 1$.
The keyword "intersection" guided me to set the two equations equal: $x+z = 1 = x^2+2y^2+z^2$ and I think that I am supposed to discover some hint here, but I cannot seem to recognize the geometric shape this equation is giving me(can't match the quadric surface formulas I know), can anyone provide a hint?

Answer 1

Solve for $z$ in the plane equation:

$$x+z=1 \implies z=1-x$$

Substitute this into the equation for the other surface:

$$\begin{align*}z=1-x\implies x^2+2y^2+(1-x)^2&=1 \\ x^2 - x + y^2 &=0 \\ \left(x-\frac12\right)^2+y^2&=\frac14 \end{align*}$$

which should look familiar.