Consider a branch of the hyperbola, \( x^{2}-2 y^{2}-2 \sqrt{2} x-4 \sqrt{2} y-6=0 \) with verte...

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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 \( \mathrm{B} \) be one of the end points of its latus rectum. If \( \mathrm{C} \) is the focus of the hyperbola nearest to the point \( \mathrm{A} \),
\( \mathrm{P} \) then the area of the triangle \( A B C \) is

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(A) \( 1-\sqrt{\frac{2}{3}} \)
(B) \( \sqrt{\frac{3}{2}}-1 \)
(C) \( 1+\sqrt{\frac{2}{3}} \)
(D) \( \sqrt{\frac{3}{2}}+1 \)
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