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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
\( \mathrm{P} \) the endpoints of its latus rectum. If \( C \) is the focus of the

W hyperbola nearest to the point \( A \), then the area of triangle \( A B C \) is
(1) \( 1-\sqrt{2 / 3} \)
(2) \( \sqrt{3 / 2}-1 \)
(3) \( 1+\sqrt{2 / 3} \)
(4) \( \sqrt{3 / 2}+1 \)
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