Let `'z'` be a comlex number and `'a'` be a real parameter such that `z^(2)+az+a^(2)=0`, then which is of the following is not true ?

Options

A. locus of `z` is a pair of straight lines
B. `|z|=|a|`
C. `arg(z)=+-(2pi)/(3)`
D. None of these

Explanation

To solve the problem, we need to analyze the quadratic equation given by:

\[ z^2 + az + a^2 = 0 \]

where \( z \) is a complex number and \( a \) is a real parameter. We will use the quadratic formula to find the roots of this equation.

### Step 1: Identify the coefficients
In the quadratic equation \( z^2 + az + a^2 = 0 \), we identify the coefficients:
- \( A = 1 \) (coefficient of \( z^2 \))
- \( B = a \) (coefficient of \( z \))
- \( C = a^2 \) (constant term)

### Step 2: Apply the quadratic formula
The quadratic formula for finding the roots of the equation \( Ax^2 + Bx + C = 0 \) is given by:

\[ z = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Substituting the values of \( A \), \( B \), and \( C \):

\[ z = \frac{-a \pm \sqrt{a^2 - 4(1)(a^2)}}{2(1)} \]

### Step 3: Simplify the expression under the square root
Now, simplify the expression under the square root:

\[ z = \frac{-a \pm \sqrt{a^2 - 4a^2}}{2} \]
\[ z = \frac{-a \pm \sqrt{-3a^2}}{2} \]
\[ z = \frac{-a \pm \sqrt{3}i a}{2} \]

### Step 4: Separate the roots
We can separate the two roots:

1. \( z_1 = \frac{-a + \sqrt{3}i a}{2} \)
2. \( z_2 = \frac{-a - \sqrt{3}i a}{2} \)

### Step 5: Factor out \( a \)
Factoring out \( a \) from both roots gives:

1. \( z_1 = a \left( \frac{-1 + \sqrt{3}i}{2} \right) \)
2. \( z_2 = a \left( \frac{-1 - \sqrt{3}i}{2} \right) \)

### Step 6: Analyze the arguments of the roots
The arguments of the complex numbers can be calculated. The roots can be expressed in polar form. The argument of \( z_1 \) and \( z_2 \) can be determined as follows:

- The argument of \( z_1 \) is \( \tan^{-1}\left(\frac{\sqrt{3}}{-1}\right) + \pi \) (since it lies in the second quadrant).
- The argument of \( z_2 \) is \( \tan^{-1}\left(\frac{-\sqrt{3}}{-1}\right) \) (since it lies in the third quadrant).

Calculating these gives:

- Argument of \( z_1 = \frac{2\pi}{3} \)
- Argument of \( z_2 = -\frac{2\pi}{3} \)

### Conclusion
From the analysis, we can conclude that the arguments of the roots are \( \frac{2\pi}{3} \) and \( -\frac{2\pi}{3} \), while the argument of the real parameter \( a \) is \( 0 \).

Thus, the statement that is not true among the options provided would be the one that incorrectly relates the argument of \( z \) to that of \( a \).