Complex number z lies on the curve `S-= arg(z+3)/(z+3i) =-(pi)/(4)`

Options

A. `{:(,,"a","b","c","d"),(,"(1)","p","q","p","r"):}`
B. `{:(,,"a","b","c","d"),(,"(2)","s","r","q","p"):}`
C. `{:(,,"a","b","c","d"),(,"(3)","q","p","q","r"):}`
D. `{:(,,"a","b","c","d"),(,"(4)","s","p","q","r"):}`

Answer

To solve the problem where the complex number \( z \) lies on the curve defined by the equation

\[ S = \frac{\arg(z + 3)}{z + 3i} = -\frac{\pi}{4}, \]

we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation in a more manageable form. We know that

\[ \arg(z + 3) = -\frac{\pi}{4} \cdot (z + 3i). \]

### Step 2: Express \( z \) in terms of \( x \) and \( y \)
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Then we can express \( z + 3 \) as:

\[ z + 3 = (x + 3) + yi. \]

### Step 3: Find the argument
The argument of a complex number \( a + bi \) is given by

\[ \arg(a + bi) = \tan^{-1}\left(\frac{b}{a}\right). \]

Thus, we have:

\[ \arg(z + 3) = \tan^{-1}\left(\frac{y}{x + 3}\right). \]

### Step 4: Substitute into the equation
Substituting this into our equation gives us:

\[ \frac{\tan^{-1}\left(\frac{y}{x + 3}\right)}{(x + 3) + yi} = -\frac{\pi}{4}. \]

### Step 5: Multiply both sides by \( (x + 3) + yi \)
To eliminate the denominator, we multiply both sides by \( (x + 3) + yi \):

\[ \tan^{-1}\left(\frac{y}{x + 3}\right) = -\frac{\pi}{4} \cdot ((x + 3) + yi). \]

### Step 6: Analyze the equation
From the equation above, we can analyze the real and imaginary parts. The left-hand side is a real number, while the right-hand side is a complex number. For the equality to hold, the imaginary part must be zero. Thus:

\[ -\frac{\pi}{4} \cdot y = 0 \implies y = 0. \]

### Step 7: Substitute \( y = 0 \)
Now substituting \( y = 0 \) back into our expression for \( z \):

\[ z = x + 0i = x. \]

### Step 8: Solve for \( x \)
Now we need to find the values of \( x \) that satisfy the original equation. Since \( y = 0 \), we have:

\[ \arg(z + 3) = \arg(x + 3) = \tan^{-1}\left(\frac{0}{x + 3}\right) = 0. \]

Thus, we need to find \( x \) such that:

\[ 0 = -\frac{\pi}{4} \cdot (x + 3). \]

This implies:

\[ x + 3 = 0 \implies x = -3. \]

### Final Result
Thus, the complex number \( z \) that lies on the curve is:

\[ z = -3. \]