In Argand diagram, O, P, Q represent the origin, z and z+ iz respectively then `angle OPQ`=

Options

A. `pi/4`
B. `pi/3`
C. `pi/2`
D. `(2pi)/3`

Answer

To find the angle \( \angle OPQ \) in the Argand diagram where \( O \) is the origin, \( P \) is represented by the complex number \( z \), and \( Q \) is represented by \( z + iz \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( O \) be the origin, represented as \( O(0, 0) \).
- Let \( P \) be the point represented by the complex number \( z \), which can be expressed as \( P(a, b) \) where \( z = a + bi \).
- Let \( Q \) be the point represented by \( z + iz \). This can be simplified as follows:
\[ Q = z + iz = a + bi + i(a + bi) = a + bi + ai - b = (a - b) + (a + b)i \]
- Thus, \( Q \) can be represented as \( Q(a - b, a + b) \).

2. **Determine the Vectors**:
- The vector \( \overrightarrow{OP} = P - O = (a, b) \).
- The vector \( \overrightarrow{OQ} = Q - O = (a - b, a + b) \).

3. **Calculate the Angle**:
- The angle \( \angle OPQ \) can be found using the dot product formula:
\[ \cos(\theta) = \frac{\overrightarrow{OP} \cdot \overrightarrow{OQ}}{|\overrightarrow{OP}| |\overrightarrow{OQ}|} \]
- Calculate the dot product \( \overrightarrow{OP} \cdot \overrightarrow{OQ} \):
\[ \overrightarrow{OP} \cdot \overrightarrow{OQ} = a(a - b) + b(a + b) = a^2 - ab + ab + b^2 = a^2 + b^2 \]
- Calculate the magnitudes:
\[ |\overrightarrow{OP}| = \sqrt{a^2 + b^2} \]
\[ |\overrightarrow{OQ}| = \sqrt{(a - b)^2 + (a + b)^2} = \sqrt{(a^2 - 2ab + b^2) + (a^2 + 2ab + b^2)} = \sqrt{2a^2 + 2b^2} = \sqrt{2} \sqrt{a^2 + b^2} \]

4. **Substitute into the Cosine Formula**:
- Substitute the values into the cosine formula:
\[ \cos(\theta) = \frac{a^2 + b^2}{\sqrt{a^2 + b^2} \cdot \sqrt{2} \sqrt{a^2 + b^2}} = \frac{1}{\sqrt{2}} \]
- This means that \( \theta = \frac{\pi}{4} \) radians.

5. **Determine the Angle \( \angle OPQ \)**:
- Since \( Q \) is obtained by rotating \( P \) by \( 90^\circ \) (as multiplying by \( i \) represents a \( 90^\circ \) rotation), we can conclude:
\[ \angle OPQ = 90^\circ \]

### Final Answer:
\[ \angle OPQ = 90^\circ \]