Step-by-Step Solution
Key Concept: For a hyperbola, asymptotes are found by replacing the constant term with zero in the standard form equation. The asymptote equation is obtained by setting the second-degree terms equal to zero: if hyperbola is ax² + by² + 2hxy + 2gx + 2fy + c = 0, then asymptotes satisfy ax² + by² + 2hxy + 2gx + 2fy = 0.
The parametric equation of the tangent is $\frac{x-1}{-1/\sqrt{2}} = \frac{y-1}{1/\sqrt{2}} = \pm 3\sqrt{2}$. The asymptotes of the hyperbola are given by $2y+x=0$ and $y-4 = -\frac{1}{2}(x+2)$, which simplify to $2y+x=0$ and $2y+x=4$. The vertices are $A=(4,-2)$ and $B=(-2,-4)$.
Correct Answer: 1