Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
Given $|3z_1 - 2z_2 - 4|^2 = |3z_1 - 1|^2 + |2z_2 + 3|^2$, $z_2 = -\frac{3}{2}$. If cube roots of $w = \frac{3z_1 - 1}{2z_2 + 3}$ are $w_1, w_2, w_3$ where $(\arg w_1 < \arg w_2 < \arg w_3)$, then the value of $\frac{w_2}{w_1 w_3}$ is ____.
Step-by-Step Solution
Key Concept: Subtract degenerate cases (collinear points) systematically from the total to count valid triangles.
Total triangles from $K$ vertices is $\binom{K}{3}$. Triangles with all three sides on polygon edges: 0. Triangles with exactly two sides on polygon edges: $k$. Triangles with exactly one side on polygon edges involve choosing one edge from $K$ sides, then one vertex not adjacent to it: $K\binom{K-4}{1} - K(k-1)\frac{(k-2)}{6} = \frac{k(k-4)(k-5)}{6} = 50$.
Correct Answer: 1