If `a gt 0 and z|z| + az + 3i = 0`, then z is

Options

A. 0
B. purely imaginary
C. a positive real number
D. a negative real number

Answer

To solve the equation \( z |z| + az + 3i = 0 \) where \( a > 0 \), we start by expressing \( z \) in terms of its real and imaginary parts. Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 1: Substitute \( z \) into the equation
We have:
\[ |z| = \sqrt{x^2 + y^2} \]
Thus, the equation becomes:
\[ (x + iy) \sqrt{x^2 + y^2} + a(x + iy) + 3i = 0 \]

### Step 2: Separate real and imaginary parts
Expanding the equation gives us:
\[ x \sqrt{x^2 + y^2} + a x + 3i + i y \sqrt{x^2 + y^2} + aiy = 0 \]
This can be separated into real and imaginary parts:
- Real part: \( x \sqrt{x^2 + y^2} + ax = 0 \)
- Imaginary part: \( y \sqrt{x^2 + y^2} + ay + 3 = 0 \)

### Step 3: Analyze the real part
From the real part:
\[ x(\sqrt{x^2 + y^2} + a) = 0 \]
This gives us two cases:
1. \( x = 0 \)
2. \( \sqrt{x^2 + y^2} + a = 0 \) (not possible since \( a > 0 \))

Thus, we conclude:
\[ x = 0 \]

### Step 4: Analyze the imaginary part
Substituting \( x = 0 \) into the imaginary part:
\[ y \sqrt{0^2 + y^2} + ay + 3 = 0 \]
This simplifies to:
\[ y |y| + ay + 3 = 0 \]

### Step 5: Consider cases for \( y \)
1. **If \( y \geq 0 \)**:
\[ y^2 + ay + 3 = 0 \]
The discriminant \( D = a^2 - 4 \cdot 1 \cdot 3 = a^2 - 12 \). For \( D < 0 \) (since \( a > 0 \)), there are no real solutions for \( y \geq 0 \).

2. **If \( y < 0 \)**:
\[ -y^2 + ay + 3 = 0 \implies y^2 - ay - 3 = 0 \]
The discriminant \( D = a^2 + 4 \cdot 1 \cdot 3 = a^2 + 12 \) is always positive, indicating two real solutions.

### Conclusion
Since \( x = 0 \) and \( y < 0 \), we find that \( z \) must be purely imaginary and negative. Therefore, the solution is:
\[ z = iy \quad \text{where } y < 0 \]

### Final Answer
Thus, \( z \) is purely imaginary.