Let $AB$ and $CD$ be parallel chords of the circle $|z| = r$. If $z_1, z_2, z_3$ and $z_4$ represent $A, B, C$ and $D$, respectively, and $z_1 z_2 = kz_3 z_4$ then $\frac{z_3 z_4}{4}$ equals ____.
Step-by-Step Solution
Key Concept: Each element outside the intersection has three independent choices: belong to $A$ only, $B$ only, or neither.
Given $A \cap B = \{2, 3, 5, 7\}$, the remaining elements $\{1, 4, 6, 8, 9, 10\}$ can belong to either set $A$, set $B$, or neither. Each of the 6 elements has 3 options, giving $3^6$ total configurations. Since $A ≠ B$, we exclude the case where both are identical, yielding $3^6 - 1 = 729 - 1 = 728$ ordered pairs.
Correct Answer: 18.75