Area of Triangle Formed by Hyperbola Vertex Focus and Latus Rectum

To find the area of the triangle $ABC$, we first rewrite the equation of the given hyperbola in standard form:

$x^2 - 2y^2 - 2\sqrt{2}x - 4\sqrt{2}y - 6 = 0$

Group the $x$ and $y$ terms and complete the squares:
$(x^2 - 2\sqrt{2}x) - 2(y^2 + 2\sqrt{2}y) = 6$
$\left((x - \sqrt{2})^2 - 2\right) - 2\left((y + \sqrt{2})^2 - 2\right) = 6$
$(x - \sqrt{2})^2 - 2 - 2(y + \sqrt{2})^2 + 4 = 6$
$(x - \sqrt{2})^2 - 2(y + \sqrt{2})^2 = 4$

Dividing both sides by $4$, we get the standard form of the hyperbola:
$\frac{(x - \sqrt{2})^2}{4} - \frac{(y + \sqrt{2})^2}{2} = 1$

Let $X = x - \sqrt{2}$ and $Y = y + \sqrt{2}$. The equation becomes:
$\frac{X^2}{a^2} - \frac{Y^2}{b^2} = 1$

where:
$a^2 = 4 \implies a = 2$
$b^2 = 2 \implies b = \sqrt{2}$

The eccentricity $e$ of the hyperbola is:
$e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{2}{4}} = \sqrt{\frac{3}{2}}$

In the $(X, Y)$ coordinate system, the coordinates of the key points are:

1. Vertex ($A$): Lies on the transverse axis at $(a, 0) = (2, 0)$.
2. Focus ($C$): The focus nearest to $A$ is located at $(ae, 0) = \left(2\sqrt{\frac{3}{2}}, 0\right)$.
3. Endpoint of Latus Rectum ($B$): The latus rectum passes through the focus $C$ perpendicular to the transverse axis. Thus, $B$ has coordinates $\left(ae, \frac{b^2}{a}\right) = \left(2\sqrt{\frac{3}{2}}, 1\right)$.

Since $A$ and $C$ both lie on the $X$-axis ($Y = 0$), and $B$ and $C$ share the same $X$-coordinate ($X = ae$), the line segment $BC$ is perpendicular to $AC$. Therefore, $\triangle ABC$ is a right-angled triangle with the right angle at $C$.

The length of the base $AC$ is:
$AC = ae - a = a(e - 1) = 2\left(\sqrt{\frac{3}{2}} - 1\right)$

The length of the height $BC$ is:
$BC = \frac{b^2}{a} = \frac{2}{2} = 1$

The area of $\triangle ABC$ is given by:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AC \times BC$
$\text{Area} = \frac{1}{2} \times 2\left(\sqrt{\frac{3}{2}} - 1\right) \times 1 = \sqrt{\frac{3}{2}} - 1$

Thus, the correct option is B.