Asymptotes of a Rectangular Hyperbola

The equation of the hyperbola is given by $$xy - 3x - 2y = 0$$. This can be factored by grouping terms together:

$$x(y - 3) - 2(y - 3) - 6 = 0 \implies (x - 2)(y - 3) = 6$$

The equations of the asymptotes are obtained by setting the constant product term to zero:

$$(x - 2)(y - 3) = 0 \implies x = 2 \text{ and } y = 3$$

The three lines forming the triangle are:

1) $$x = 2$$

2) $$y = 3$$

3) $$x + y = 2$$

Find the three intersection vertices of these lines:

Intersection of $$x = 2$$ and $$y = 3$$ is $$A(2, 3)$$.

Intersection of $$x = 2$$ and $$x + y = 2$$ is $$B(2, 0)$$.

Intersection of $$y = 3$$ and $$x + y = 2$$ is $$C(-1, 3)$$.

The triangle is right-angled at vertex $$A(2, 3)$$.

Length of the vertical base $$AB = \sqrt{(2-2)^2 + (3-0)^2} = 3$$.

Length of the horizontal height $$AC = \sqrt{(2 - (-1))^2 + (3-3)^2} = 3$$.

The area of the right-angled triangle is:

$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 3 = \frac{9}{2}$$