If `a, b in C and z` is a non zero complex number then the root of the equation `az^3 + bz^2 + bar(b) z + bar(a)=0` lie on

Options

A. the circle `x^2 + y^2 = |a|^2`
B. the circle `x^2 + y^2=1`
C. the circle `x^2 + y^2 = |a|^2 +|b|^2`
D. the circle `x^2 + y^2 = |a|^2 - |b|^2`

Answer

To solve the equation \( az^3 + bz^2 + \bar{b} z + \bar{a} = 0 \) and determine where its roots lie, we can follow these steps:

### Step 1: Identify the roots
Let the roots of the polynomial be \( z_1, z_2, z_3 \). According to Vieta's formulas, we can express the relationships between the coefficients and the roots.

### Step 2: Use Vieta's Formulas
From Vieta's formulas, we have:
- The sum of the roots:
\[ z_1 + z_2 + z_3 = -\frac{b}{a} \]
- The sum of the products of the roots taken two at a time:
\[ z_1z_2 + z_2z_3 + z_3z_1 = \frac{\bar{b}}{a} \]
- The product of the roots:
\[ z_1 z_2 z_3 = -\frac{\bar{a}}{a} \]

### Step 3: Analyze the modulus of the roots
To analyze where the roots lie, we will take the modulus of the relationships we derived from Vieta's formulas.

1. Taking the modulus of the sum of the roots:
\[ |z_1 + z_2 + z_3| = \left| -\frac{b}{a} \right| = \frac{|b|}{|a|} \]

2. Taking the modulus of the sum of the products of the roots:
\[ |z_1 z_2 + z_2 z_3 + z_3 z_1| = \left| \frac{\bar{b}}{a} \right| = \frac{|b|}{|a|} \]

3. Taking the modulus of the product of the roots:
\[ |z_1 z_2 z_3| = \left| -\frac{\bar{a}}{a} \right| = \frac{|\bar{a}|}{|a|} = \frac{|a|}{|a|} = 1 \]

### Step 4: Relate the moduli
From the above, we can derive:
- The modulus of the sum of the roots and the sum of the products of the roots gives us a relation that can be simplified.
- Since \( |z_1 z_2 z_3| = 1 \), we can conclude that the product of the roots has a modulus of 1.

### Step 5: Conclude the location of the roots
Given that the product of the roots has a modulus of 1, and the relationships derived from Vieta’s formulas suggest that the roots are symmetric in nature, we can conclude that:
- The roots \( z_1, z_2, z_3 \) lie on the unit circle in the complex plane, which is defined by \( |z| = 1 \).

### Final Conclusion
Thus, the roots of the equation \( az^3 + bz^2 + \bar{b} z + \bar{a} = 0 \) lie on the circle of radius 1 centered at the origin in the complex plane.

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