Consider a branch of the hyperbola
$$x^2 - 2y^2 - 2\sqrt{2}x - 4\sqrt{2}y - 6 = 0$$
with vertex at the point A. Let B be oneof the endpoints of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is

A

$$1 - \sqrt{\frac{2}{3}}$$

B

$$\sqrt{\frac{3}{2}} - 1$$

C

$$1 + \sqrt{\frac{2}{3}}$$

D

$$\sqrt{\frac{3}{2}} + 1$$