Let a, b, c be distinct complex numbers with |a| = |b| = |c| = 1. Let z1 and z2 be the roots of az² + bz + c = 0, with |z1| = 1. Points P and Q represent z1 and z2 in the complex plane, O is the origin, and angle POQ = theta. Which of the following is/are correct?

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theta = 2*pi/3. z1*z2 = c/a (|z1*z2|=1 so |z2|=1). z1+z2 = -b/a, |z1+z2| = |b/a| = 1. Since |z1|=|z2|=1 and |z1+z2|=1, we have a triangle with sides |z1|=1, |z2|=1, |z1+z2|=1 — equilateral-like. Actually |z1+z2|² = |z1|² + |z2|² + 2*Re(z1*conj(z2)) = 1+1+2cos(theta) = 1. So 2+2cos(theta)=1 => cos(theta)=-1/2 => theta=2pi/3. Also |z1-z2|² = 2-2cos(theta) = 2+1 = 3, so PQ = sqrt(3). And |z1+z2|=1 is confirmed. So options B, C, D are correct (theta=2pi/3, PQ=sqrt(3), |z1+z2|=1). For b²=ac: (z1+z2)² = b²/a² and z1*z2=c/a. b²/a² = (-b/a)² = b²/a²; and (z1+z2)² = z1²+2z1z2+z2². Also b² = ac means b²/a² = c/a = z1*z2. This would require (z1+z2)² = 4*z1*z2 i.e. (z1-z2)²=0, so z1=z2. But they need no

Let a, b, c be distinct complex numbers with |a| = |b| = |c| = 1. Let z1 and z2 be the roots of az² + bz + c = 0, with |z1| = 1. Points P and Q represent z1 and z2 in the complex plane, O is the origin, and angle POQ = theta. Which of the following is/are correct?

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