Let `Z_1` and `Z_2` are two non-zero complex number such that `|Z_1+Z_2|=|Z_1|=|Z_2|`, then `Z_1/Z_2` may be :

Options

A. `1 + omega`
B. `1+ omega^2`
C. `omega`
D. `omega^2`

Answer

To solve the problem, we need to analyze the conditions given for the complex numbers \( Z_1 \) and \( Z_2 \). We know that:

1. \( |Z_1 + Z_2| = |Z_1| \)
2. \( |Z_1 + Z_2| = |Z_2| \)

From these conditions, we can derive the possible values for \( \frac{Z_1}{Z_2} \).

### Step-by-Step Solution:

**Step 1: Understanding the Conditions**

Given \( |Z_1 + Z_2| = |Z_1| = |Z_2| \), we can denote \( |Z_1| = |Z_2| = r \) for some positive real number \( r \).

**Hint:** Recognize that the magnitudes of \( Z_1 \) and \( Z_2 \) are equal.

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**Step 2: Using the Triangle Inequality**

From the triangle inequality, we know that:

\[ |Z_1 + Z_2| \leq |Z_1| + |Z_2| = r + r = 2r \]

However, since \( |Z_1 + Z_2| = r \), we have:

\[ r \leq 2r \]

This is always true, but we also know from the equality condition of the triangle inequality that \( Z_1 \) and \( Z_2 \) must be in the same direction in the complex plane.

**Hint:** Consider the implications of the triangle equality.

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**Step 3: Expressing \( Z_1 \) and \( Z_2 \)**

Let’s express \( Z_1 \) and \( Z_2 \) in terms of a common angle. We can set:

\[ Z_1 = r e^{i\theta}, \quad Z_2 = r e^{i\phi} \]

Since \( |Z_1| = |Z_2| = r \).

**Hint:** Use polar form to express complex numbers.

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**Step 4: Analyzing the Condition \( |Z_1 + Z_2| = r \)**

Now we have:

\[ |Z_1 + Z_2| = |r e^{i\theta} + r e^{i\phi}| = r |e^{i\theta} + e^{i\phi}| \]

Setting this equal to \( r \):

\[ |e^{i\theta} + e^{i\phi}| = 1 \]

This implies that \( e^{i\theta} + e^{i\phi} \) must lie on the unit circle.

**Hint:** Consider the geometric interpretation of complex numbers on the unit circle.

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**Step 5: Finding the Ratio \( \frac{Z_1}{Z_2} \)**

Now, we can find the ratio:

\[ \frac{Z_1}{Z_2} = \frac{r e^{i\theta}}{r e^{i\phi}} = e^{i(\theta - \phi)} \]

Since \( |e^{i\theta} + e^{i\phi}| = 1 \), we can conclude that \( \theta - \phi \) can take specific values that correspond to the angles of the cube roots of unity.

**Hint:** Think about the angles that satisfy the conditions derived.

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**Step 6: Possible Values of \( \frac{Z_1}{Z_2} \)**

The possible values of \( \frac{Z_1}{Z_2} \) can be determined by the cube roots of unity:

1. \( \omega = e^{2\pi i / 3} \)
2. \( \omega^2 = e^{-2\pi i / 3} \)

Thus, \( \frac{Z_1}{Z_2} \) can be either \( \omega \) or \( \omega^2 \).

**Hint:** Recall the properties of cube roots of unity.

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### Final Answer:

The values that \( \frac{Z_1}{Z_2} \) may take are \( \omega \) and \( \omega^2 \), which correspond to option 3 and option 4.