The range of m for which the line `z(1-mi) - bar(z) (1+mi)=0` divides two chords drawn from point `z_1 = sqrt(3)+i` to the curve `|z|=2` in the ratio `1:2` is

Options

A. `( 0, sqrt3)`
B. `(1, 2)`
C. `( (1)/(sqrt3) , 1)`
D. None of these

Answer

To solve the problem, we need to find the range of \( m \) for which the line \( z(1 - mi) - \bar{z}(1 + mi) = 0 \) divides two chords drawn from the point \( z_1 = \sqrt{3} + i \) to the curve \( |z| = 2 \) in the ratio \( 1:2 \).

### Step 1: Simplifying the line equation
The line equation can be rewritten as:
\[ z(1 - mi) = \bar{z}(1 + mi) \]
Let \( z = x + iy \) and \( \bar{z} = x - iy \). Substituting these into the equation gives:
\[ (x + iy)(1 - mi) = (x - iy)(1 + mi) \]
Expanding both sides:
\[ x + iy - mx i - my = x + mxy - iy - mxy \]
This simplifies to:
\[ x - my + iy - mx i = x + mxy - iy - mxy \]
After cancelling terms, we can rearrange to find:
\[ y = mx \]
This represents the line through the origin with slope \( m \).

### Step 2: Finding the intersection points with the circle
The circle defined by \( |z| = 2 \) can be expressed as:
\[ x^2 + y^2 = 4 \]
Substituting \( y = mx \) into the circle's equation gives:
\[ x^2 + (mx)^2 = 4 \implies x^2(1 + m^2) = 4 \implies x^2 = \frac{4}{1 + m^2} \]
Thus, we have:
\[ x = \pm \frac{2}{\sqrt{1 + m^2}} \]
Substituting back to find \( y \):
\[ y = mx = \pm \frac{2m}{\sqrt{1 + m^2}} \]
The intersection points are:
\[ P_1 = \left( \frac{2}{\sqrt{1 + m^2}}, \frac{2m}{\sqrt{1 + m^2}} \right) \quad \text{and} \quad P_2 = \left( -\frac{2}{\sqrt{1 + m^2}}, -\frac{2m}{\sqrt{1 + m^2}} \right) \]

### Step 3: Using the section formula
The point \( z_1 = \sqrt{3} + i \) divides the chord \( P_1P_2 \) in the ratio \( 1:2 \). By the section formula, the coordinates of the point dividing the segment in the ratio \( 1:2 \) are:
\[ \left( \frac{2(-\frac{2}{\sqrt{1 + m^2}}) + 1(\frac{2}{\sqrt{1 + m^2}})}{1 + 2}, \frac{2(-\frac{2m}{\sqrt{1 + m^2}}) + 1(\frac{2m}{\sqrt{1 + m^2}})}{1 + 2} \right) \]
This simplifies to:
\[ \left( \frac{-\frac{4}{\sqrt{1 + m^2}} + \frac{2}{\sqrt{1 + m^2}}}{3}, \frac{-\frac{4m}{\sqrt{1 + m^2}} + \frac{2m}{\sqrt{1 + m^2}}}{3} \right) = \left( \frac{-2}{3\sqrt{1 + m^2}}, \frac{-2m}{3\sqrt{1 + m^2}} \right) \]

### Step 4: Setting the coordinates equal to \( z_1 \)
Setting these coordinates equal to \( z_1 = \sqrt{3} + i \):
\[ \frac{-2}{3\sqrt{1 + m^2}} = \sqrt{3} \quad \text{and} \quad \frac{-2m}{3\sqrt{1 + m^2}} = 1 \]

From the first equation:
\[ -2 = 3\sqrt{3}\sqrt{1 + m^2} \implies 1 + m^2 = \frac{4}{27} \implies m^2 = \frac{4}{27} - 1 = \frac{-23}{27} \quad \text{(not possible)} \]

From the second equation:
\[ -2m = 3\sqrt{1 + m^2} \implies 4m^2 = 9(1 + m^2) \implies 4m^2 = 9 + 9m^2 \implies 5m^2 = -9 \quad \text{(not possible)} \]

### Conclusion
Since both conditions lead to contradictions, we conclude that the line divides the chords in the ratio \( 1:2 \) for all values of \( m \) except where the conditions lead to non-real solutions.

### Final Result
The range of \( m \) for which the line divides the chords in the ratio \( 1:2 \) is:
\[ m \in \mathbb{R} \quad \text{(all real numbers)} \]