Let `z_1 and z_2` be two non - zero complex numbers such that `z_1/z_2+z_2/z_1=1` then the origin and points represented by `z_1 and z_2`

Options

A. `z_(1),z_(2)` are collinear
B. `z_(1),z_(2)` are the origin from a right angled triangle
C. `z_(1),z_(2)` and the origin form an equilateral triangle
D. none of these

Answer

To solve the problem, we need to analyze the given equation involving the complex numbers \( z_1 \) and \( z_2 \):

Given:
\[ \frac{z_1}{z_2} + \frac{z_2}{z_1} = 1 \]

### Step 1: Let \( z = \frac{z_1}{z_2} \)
We can rewrite the equation as:
\[ z + \frac{1}{z} = 1 \]

### Step 2: Multiply both sides by \( z \)
This gives us:
\[ z^2 + 1 = z \]

### Step 3: Rearrange the equation
Rearranging the equation, we have:
\[ z^2 - z + 1 = 0 \]

### Step 4: Use the quadratic formula
We can solve this quadratic equation using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -1, c = 1 \).

Substituting the values:
\[ z = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]

### Step 5: Simplify the expression
This simplifies to:
\[ z = \frac{1 \pm i\sqrt{3}}{2} \]

### Step 6: Find the modulus of \( z \)
Now, we compute the modulus of \( z \):
\[ |z| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

### Step 7: Relate back to \( z_1 \) and \( z_2 \)
Since \( z = \frac{z_1}{z_2} \) and \( |z| = 1 \), it follows that:
\[ |z_1| = |z_2| \]

### Step 8: Analyze the triangle formed by the origin and points represented by \( z_1 \) and \( z_2 \)
Let \( O \) be the origin, \( A \) be the point represented by \( z_1 \), and \( B \) be the point represented by \( z_2 \). Since \( |z_1| = |z_2| \), we have:
\[ OA = OB \]

### Step 9: Find the distance \( AB \)
We also know that:
\[ AB = |z_2 - z_1| = |z_2(1 - z)| = |z_2| |1 - z| \]
Since \( |z_2| = |z_1| = r \) (say), we need to find \( |1 - z| \):
\[ |1 - z| = |1 - \frac{1 \pm i\sqrt{3}}{2}| = |1 - \frac{1}{2} \mp \frac{i\sqrt{3}}{2}| = |\frac{1}{2} \mp \frac{i\sqrt{3}}{2}| \]
Calculating this modulus:
\[ |1 - z| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

### Step 10: Conclude the triangle properties
Thus, we find:
\[ OA = OB = AB \]
This implies that the points \( O, A, \) and \( B \) form an equilateral triangle.

### Final Answer
The origin and the points represented by \( z_1 \) and \( z_2 \) form an equilateral triangle.

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