Answer
It does not matter which curve is chosen.
Work Step by Step
When $F(x,y)=Ai+Bj$ is a conservative field, then throughout the domain $D$, we get
$\dfrac{\partial A}{\partial y}=\dfrac{\partial B}{\partial x}$
$a$ and $b$ are the first-order partial derivatives on the domain $D$.
$\int_C \overrightarrow{F} \cdot \overrightarrow{dr}$ is independent of path if and only if $\int_C \overrightarrow{F} \cdot \overrightarrow{dr}=0$ for every closed curve $C$.
Then, we can find that our 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.This implies that the curve has the same initial
and final point with respect to its vector function.
Hence, it has been noticed that any curve $C$ that is closed will admit a minimal work equal to $0$ in the field $\overrightarrow{F}$. Thus, it does not matter which curve is chosen.
