Let `a`, `b`, `c` be distinct complex numbers with `|a|=|b|=|c|=1` and `z_(1)`, `z_(2)` be the roots of the equation `az^(2)+bz+c=0` with `|z_(1)|=1`. Let `P` and `Q` represent the complex numbers `z_(1)` and `z_(2)` in the Argand plane with `/_POQ=theta`, `o^(@) lt 180^(@)` (where `O` being the origin).Then

Options

A. `b^(2)=ac`, `theta=(2pi)/(3)`
B. `theta=(2pi)/(3)`,`PQ=sqrt(3)`
C. `PQ=2sqrt(3)`, `b^(2)=ac`
D. `theta=(pi)/(3)`, `b^(2)=ac`

Explanation

To solve the problem step by step, we will analyze the given conditions and use properties of complex numbers.

### Step 1: Understanding the roots of the quadratic equation
We have the quadratic equation:
\[ az^2 + bz + c = 0 \]
with roots \( z_1 \) and \( z_2 \). According to Vieta's formulas, the sum and product of the roots can be expressed as:
\[ z_1 + z_2 = -\frac{b}{a} \]
\[ z_1 z_2 = \frac{c}{a} \]

### Step 2: Applying the modulus condition
Given that \( |a| = |b| = |c| = 1 \), we can conclude:
\[ |z_1 + z_2| = \left| -\frac{b}{a} \right| = 1 \]
\[ |z_1 z_2| = \left| \frac{c}{a} \right| = 1 \]

### Step 3: Using the modulus of the roots
Since \( |z_1| = 1 \) and \( |z_2| = 1 \), we can write:
\[ |z_1 + z_2| = |z_1| + |z_2| = 1 + 1 = 2 \]
However, we know from the problem statement that \( |z_1 + z_2| = 1 \). This leads us to conclude that \( z_1 \) and \( z_2 \) must be positioned such that they form a specific angle in the Argand plane.

### Step 4: Squaring the sum of the roots
We can square the modulus condition:
\[ |z_1 + z_2|^2 = 1 \]
Expanding this gives:
\[ |z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2}) = 1 \]
Since \( |z_1| = |z_2| = 1 \), we have:
\[ 1 + 1 + 2 \text{Re}(z_1 \overline{z_2}) = 1 \]
This simplifies to:
\[ 2 + 2 \text{Re}(z_1 \overline{z_2}) = 1 \]
Thus:
\[ \text{Re}(z_1 \overline{z_2}) = -\frac{1}{2} \]

### Step 5: Relating the angle
Since \( z_2 = z_1 e^{i\theta} \), we can express the real part:
\[ \text{Re}(z_1 \overline{z_2}) = \text{Re}(z_1 \cdot z_1^{-1} e^{-i\theta}) = \text{Re}(e^{-i\theta}) = \cos(\theta) \]
Thus, we have:
\[ \cos(\theta) = -\frac{1}{2} \]
This implies:
\[ \theta = \frac{2\pi}{3} \text{ or } \theta = \frac{4\pi}{3} \]
Since \( \theta < 180^\circ \), we take:
\[ \theta = \frac{2\pi}{3} \]

### Step 6: Finding the value of \( pq \)
To find \( pq \), we need to calculate \( |z_1 - z_2| \):
\[ |z_1 - z_2| = |z_1 - z_1 e^{i\theta}| = |z_1(1 - e^{i\theta})| = |1 - e^{i\theta}| \]
Using the formula for the modulus of a complex number:
\[ |1 - e^{i\theta}| = \sqrt{(1 - \cos(\theta))^2 + \sin^2(\theta)} \]
Substituting \( \cos(\theta) = -\frac{1}{2} \) and \( \sin(\theta) = \sqrt{1 - \left(-\frac{1}{2}\right)^2} = \frac{\sqrt{3}}{2} \):
\[ |1 - e^{i\theta}| = \sqrt{\left(1 + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\left(\frac{3}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} \]

### Final Result
Thus, the value of \( pq \) is:
\[ pq = \sqrt{3} \]