$w_1, w_2$ be roots of $(a+\bar{c})z^2 + (b+\bar{b})z + (\bar{a}+c) = 0$. If $|z_1| < |z_2| < 1$, then
Step-by-Step Solution
Key Concept: The quadratic equation $(a+\bar{c})z^2 + (b+\bar{b})z + (\bar{a}+c) = 0$ has a special structure where coefficients satisfy a conjugate reciprocal relationship: if we denote the equation as $Az^2 + Bz + C = 0$, then $C = \bar{A}$, making it a self-reciprocal equation. This means if $w$ is a root, then $\frac{1}{\bar{w}}$ is also a root, implying $|w_1| \cdot |w_2| = 1$.
For a triangle with sides $a, a, b$, the triangle inequality requires $2a > b$, so $b b$ to constrain possible side lengths and count systematically by ranges.
Correct Answer: 2,4