If arg ( z-1) = arg `( z+ 3 i)` , then x -1`:` y is equal to

Options

A. `3:1`
B. `1:3`
C. `3:2`
D. `2:3`

Answer

To solve the problem where \( \arg(z - 1) = \arg(z + 3i) \), we will follow these steps:

### Step 1: Express \( z \) in terms of \( x \) and \( y \)
Let \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number \( z \).

### Step 2: Write the arguments
We need to find the arguments of \( z - 1 \) and \( z + 3i \):
- \( z - 1 = (x - 1) + yi \)
- \( z + 3i = x + (y + 3)i \)

### Step 3: Set up the argument equations
The arguments can be expressed as:
\[ \arg(z - 1) = \tan^{-1}\left(\frac{y}{x - 1}\right) \]
\[ \arg(z + 3i) = \tan^{-1}\left(\frac{y + 3}{x}\right) \]

### Step 4: Set the arguments equal
Since \( \arg(z - 1) = \arg(z + 3i) \), we have:
\[ \tan^{-1}\left(\frac{y}{x - 1}\right) = \tan^{-1}\left(\frac{y + 3}{x}\right) \]

### Step 5: Eliminate the inverse tangent
Taking the tangent of both sides gives:
\[ \frac{y}{x - 1} = \frac{y + 3}{x} \]

### Step 6: Cross-multiply
Cross-multiplying gives:
\[ y \cdot x = (y + 3)(x - 1) \]

### Step 7: Expand and simplify
Expanding the right side:
\[ yx = yx - y + 3x - 3 \]
Now, cancel \( yx \) from both sides:
\[ 0 = -y + 3x - 3 \]
Rearranging gives:
\[ y = 3x - 3 \]

### Step 8: Find the ratio \( \frac{x - 1}{y} \)
Substituting \( y \) into the ratio:
\[ \frac{x - 1}{y} = \frac{x - 1}{3x - 3} \]
This simplifies to:
\[ \frac{x - 1}{3(x - 1)} = \frac{1}{3} \]

### Final Answer
Thus, the ratio \( x - 1 : y \) is \( 1 : 3 \).

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