Let A,B,C be three collinear points which are such that AB.AC=1 and the points are represented in the Argand plane by the complex numbers, 0, `z_(1)` and `z_(2)` respectively. Then,

Options

A. `z_(1)z_(2)=1`
B. `z_(1)barz_(2)=1`
C. `|z_(1)||z_(2)|=1`
D. `z_(1)=barz_(2)|`

Answer

To solve the problem, we need to analyze the given information about the collinear points A, B, and C represented by the complex numbers 0, \( z_1 \), and \( z_2 \) respectively. We know that the product of the distances \( AB \) and \( AC \) equals 1.

Here’s the step-by-step solution:

### Step 1: Understand the distances
The points A, B, and C are represented by the complex numbers:
- A at \( 0 \)
- B at \( z_1 \)
- C at \( z_2 \)

The distances can be expressed as:
- \( AB = |z_1 - 0| = |z_1| \)
- \( AC = |z_2 - 0| = |z_2| \)

### Step 2: Set up the equation
According to the problem, we have the relationship:
\[ AB \cdot AC = 1 \]
Substituting the distances we found:
\[ |z_1| \cdot |z_2| = 1 \]

### Step 3: Rewrite the equation
We can rewrite the equation as:
\[ |z_1| \cdot |z_2| = 1 \]
This implies:
\[ |z_1| = \frac{1}{|z_2|} \]

### Step 4: Express in terms of squares
Squaring both sides gives:
\[ |z_1|^2 \cdot |z_2|^2 = 1 \]
Using the property of modulus:
\[ z_1 \cdot \overline{z_1} \cdot z_2 \cdot \overline{z_2} = 1 \]
This can be simplified to:
\[ z_1 z_2 \cdot \overline{z_1 z_2} = 1 \]

### Step 5: Conclusion
Thus, we conclude that:
\[ |z_1 z_2|^2 = 1 \]
This means:
\[ |z_1 z_2| = 1 \]