Consider a quadratic equaiton `az^(2) + bz + c=0`, where a,b,c are complex number. If equaiton has two purely imaginary roots, then which of the following is not ture.

Options

A. `abarb` is purely imaginary
B. `b barc` is purely imaginary
C. `cbara` is purely real
D. none of these

Explanation

To solve the problem, we need to analyze the conditions under which a quadratic equation with complex coefficients has purely imaginary roots. The equation is given as:

\[ az^2 + bz + c = 0 \]

where \( a, b, c \) are complex numbers. Let's denote the purely imaginary roots as \( z_1 = yi \) and \( z_2 = -yi \), where \( y \) is a real number.

### Step 1: Write the conjugate of the equation
The conjugate of the equation is:

\[ a \overline{z^2} + b \overline{z} + \overline{c} = 0 \]

### Step 2: Substitute the roots
Since \( z_1 \) and \( z_2 \) are purely imaginary, we have:

\[ \overline{z_1} = -z_1 \quad \text{and} \quad \overline{z_2} = -z_2 \]

Thus, we can express the conjugate equation as:

\[ a(-z_1^2) + b(-z_1) + \overline{c} = 0 \]

This simplifies to:

\[ -a z_1^2 - b z_1 + \overline{c} = 0 \]

### Step 3: Relate the coefficients
Since both equations have the same roots, we can equate the coefficients. From the original equation and its conjugate, we can derive the following relationships:

1. \( \frac{a}{\overline{a}} = \frac{b}{-b} \)
2. \( \frac{b}{\overline{b}} = \frac{c}{\overline{c}} \)
3. \( \frac{a}{\overline{a}} = \frac{c}{\overline{c}} \)

### Step 4: Analyze the implications
From these relationships, we can derive the following:

1. \( ab^* = -a^*b \) implies \( ab^* \) is purely imaginary.
2. \( bc^* = -b^*c \) implies \( bc^* \) is purely imaginary.
3. \( ac^* = a^*c \) implies \( ac^* \) is purely real.

### Step 5: Identify which statement is not true
Given the derived conditions, we can summarize:

- \( ab^* \) is purely imaginary.
- \( bc^* \) is purely imaginary.
- \( ac^* \) is purely real.

Thus, if we are asked to identify which of the following statements is not true, we can conclude that:

- The statement that \( ac^* \) is purely imaginary is **not true**.

### Final Answer
The statement that is not true is:

**Option D: \( ac^* \) is purely imaginary.**