Consider a branch of the hyperbola
$$x ^ { 2 } - 2 y ^ { 2 } - 2 \sqrt { 2 } x - 4 \sqrt { 2 } y - 6 = 0$$
with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $A B C$ is\\
(A) $1 - \sqrt { \frac { 2 } { 3 } }$\\
(B) $\sqrt { \frac { 3 } { 2 } } - 1$\\
(C) $1 + \sqrt { \frac { 2 } { 3 } }$\\
(D) $\sqrt { \frac { 3 } { 2 } } + 1$