In a plane, $\vec{a}$ and $\vec{b}$ are the position vectors of two points $A$ and $B$ respectively. $A$ point $P$ with position vector $\vec{r}$ moves on that plane in such a way that $|\vec{r}-\vec{a}| - |\vec{r}-\vec{b}| = c$ (where $c$ is a real constant). The locus of $P$ is a conic section whose eccentricity is:

• A
  $\frac{|\vec{a}-\vec{b}|}{c}$
• B
  $\frac{|\vec{a}+\vec{b}|}{c}$
• C
  $\frac{|\vec{a}-\vec{b}|}{2c}$
• D
  $\frac{|\vec{a}+\vec{b}|}{2c}$