If z =a + ib satisfies arg (z-1) =arg (z+3i) , then `(a-1):b`=

Options

A. `2 : 1`
B. `1 : 3`
C. `-1 : 3`
D. None of these

Answer

To solve the problem, we need to find the ratio \((a-1):b\) given that \(z = a + ib\) satisfies the equation \(\arg(z - 1) = \arg(z + 3i)\).

### Step-by-step Solution:

1. **Write the expressions for \(z - 1\) and \(z + 3i\)**:
\[ z - 1 = (a + ib) - 1 = (a - 1) + ib \]
\[ z + 3i = (a + ib) + 3i = a + i(b + 3) \]

2. **Set up the equation for the arguments**:
Since \(\arg(z - 1) = \arg(z + 3i)\), we can write:
\[ \arg((a - 1) + ib) = \arg(a + i(b + 3)) \]

3. **Use the definition of argument**:
The argument of a complex number \(x + iy\) is given by \(\tan^{-1}(\frac{y}{x})\). Therefore, we have:
\[ \tan^{-1}\left(\frac{b}{a - 1}\right) = \tan^{-1}\left(\frac{b + 3}{a}\right) \]

4. **Equate the tangents**:
Since the arguments are equal, we can equate their tangents:
\[ \frac{b}{a - 1} = \frac{b + 3}{a} \]

5. **Cross-multiply to eliminate the fractions**:
\[ b \cdot a = (a - 1)(b + 3) \]

6. **Expand the right-hand side**:
\[ ab = ab + 3a - b - 3 \]

7. **Rearrange the equation**:
\[ ab - ab + b + 3 = 3a \]
\[ b + 3 = 3a \]

8. **Isolate \(b\)**:
\[ b = 3a - 3 \]

9. **Find the ratio \((a - 1):b\)**:
Substitute \(b\) into the ratio:
\[ \frac{a - 1}{b} = \frac{a - 1}{3a - 3} \]

10. **Simplify the ratio**:
Factor out the common term in the denominator:
\[ \frac{a - 1}{3(a - 1)} = \frac{1}{3} \]

Thus, the ratio \((a - 1):b\) is:
\[ \frac{1}{3} \]

### Final Answer:
\[ (a-1):b = 1:3 \]