Complex Numbers
Modulus sum for polygon vertices
MJAT_TS1_P2
Grade 12
Question:
Let $A_1, A_2, \ldots, A_7$ be the vertices of a heptagon and $a_1, a_2, \ldots, a_7$ be the complex numbers representing them. If $|a_1| = |a_2| = \cdots = |a_7| = R$, then $\displaystyle\sum_{1 \leq i < j \leq 7} |a_i + a_j|^2$
A) is greater than $30R^2$
B) has minimum value $35R^2$
C) has its minimum value in $(25R^2,\, 45R^2)$
D) is less than $45R^2$
Step-by-Step Solution
Key Concept: Expand: $\sum_{i<j}|a_i+a_j|^2 = \sum_{i<j}(|a_i|^2 + 2\text{Re}(a_i\overline{a_j}) + |a_j|^2) = 2\binom{7}{2}R^2 + 2\text{Re}\sum_{i<j}a_i\overline{a_j}$. Note $|\sum a_i|^2 = 7R^2 + 2\text{Re}\sum_{i<j}a_i\overline{a_j} \geq 0$, so the cross-term $\geq -7R^2/2$.
$\sum_{i<j}|a_i+a_j|^2 = 6\cdot 7R^2 + \left(|\sum a_i|^2 - 7R^2\right) = 35R^2 + |\sum a_i|^2 \geq 35R^2$. Minimum $= 35R^2$ (when $\sum a_i=0$). Always $> 30R^2$ ✓. Min $35R^2 \in (25R^2, 45R^2)$ ✓. But maximum can reach $49R^2 > 45R^2$, so D is not always true. Answer: A, B, C.
Correct Answer: ABC