Problem Statement
Let $w_1, w_2, \dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that \(\sum_{k = 1}^{n} (z_k - w_k) = 0\). For the numbers $w_1 = 32 + 170i$, $w_2 = - 7 + 64i$, $w_3 = - 9 + 200i$, $w_4 = 1 + 27i$, and $w_5 = - 14 + 43i$, there is a unique mean line with $y$-intercept 3. Find the slope of this mean line.
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Solution
$$\text{Line L: } y = (mx + 3)i$$ $$\therefore z_k = x_k + (mx_k + 3)i, {1 <= k <= 5}$$ $$\sum_{k = 1}^{n} (z_k - w_k) = 0 \implies \sum_{k = 1}^{n} (w_k) = \sum_{k = 1}^{n} (z_k)$$ $$3 + 504i = (x_1 + x_2 + x_3 + x_4 + x_5) + (m(x_1 + x_2 + x_3 + x_4 + x_5) + 15)i$$ $$\therefore 3 = (x_1 + x_2 + x_3 + x_4 + x_5), 504i = (m(x_1 + x_2 + x_3 + x_4 + x_5) + 15)i$$ $$504i = (m(x_1 + x_2 + x_3 + x_4 + x_5) + 15)i$$ $$504 = 3m + 15 \implies m = 163$$ 