Answer
It does not matter which curve is chosen.
Work Step by Step
The vector field $F(x,y)=Pi+Qj$ is a conservative field throughout the domain $D$, when
$\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$
Here, $P$ and $Q$ are the first-order partial derivatives on the domain $D$.
The work integral field $\int_C \overrightarrow{F} \cdot \overrightarrow{dr}$ will be independent of the path if and only if $\int_C \overrightarrow{F} \cdot \overrightarrow{dr}=0$ for every closed curve $C$.
This implies that the work integral $W= \int_C \overrightarrow{F} \cdot \overrightarrow{dr}$ is minimized when the vector field $F$ is conservative, and when the curve $C$ is closed with the same initial and terminal point with respect to its vector function $F(x,y)$.
Hence, it does not matter which curve is chosen.
