Line integral. Conservative vector field

Question

Please help me with solution of the given problem:
Let $\gamma_1$ and $\gamma_2$ be the circles of $R^2$ of radius $r=2$, centered at the points $(0,0)$ and $(8,6)$ respectively. Assume that $\gamma_1$ is clockwise oriented and that $\gamma_2$ is counterclockwise oriented. Let $F\colon R^2\setminus \{(0,0)\} \to R^2$ be a conservative vector field of class $C^1$. Denote by $C_1$ and $C_2$ the line integrals of $F$ along $\gamma_1$ and $\gamma_2$ respectively. Decide which on of the following is true:
(a) $C_1$ $\ne$ $0$ and $C_1$ = -$C_2$
(b) $C_1$ = $C_2$
(c) $C_1$ $\ne$ $0$ and $C_1$ $\ne$ $C_2$
(d) none of the other answers is correct
(e) $C_1$ $\ne$ $0$ and $C_1$ = $C_2$

Answer 1

As far as I can see, there's nothing in the question that rules out having $F$ be identically zero, in which case $C_1$ would be zero; that immediately rules out (a), (c), and (e). And distinguishing between (b) and (d) just requires you to recall whether the theorem on conservative vector fields holds when there's a discontinuity in the vector field.