Let $\alpha$ and $\beta$ be two complex numbers satisfying $|\alpha + i| = 1$ and $|\beta - 2 - 3i| = 6$. Then the value of $6|\alpha|_{\max} - |\beta|_{\max}$ is ____.
Step-by-Step Solution
Key Concept: Subtract intersection points lost to concurrency and parallelism rather than try to count valid intersections directly.
From 5 points, we can draw $\binom{5}{2} = 10$ lines. Each point has 6 perpendiculars from other 4 points, giving $5 \times 6 = 30$ total perpendiculars. Maximum intersections = $\binom{30}{2} = 435$. Subtract: 30 points lost (3 parallel perpendiculars from one line), 20 points (orthocenter concurrency per triangle), 70 points (concurrent perpendiculars from each original point). Total = $435 - 30 - 20 - 70 = 315$ distinct intersection points.
Correct Answer: 4.88