If `z_1 = –3 + 5 i, z_2 = –5 – 3 i` and z is a complex number lying on the line segment joining `z_1 & z_2`, then arg(z) can be

Options

A. `-(3pi)/(4)`
B. `-p/4`
C. `pi/6`
D. `(5pi)/6`

Answer

To find the argument of the complex number \( z \) that lies on the line segment joining \( z_1 = -3 + 5i \) and \( z_2 = -5 - 3i \), we can follow these steps:

### Step 1: Identify the Points in the Complex Plane
We have two complex numbers:
- \( z_1 = -3 + 5i \) corresponds to the point (-3, 5) in the complex plane.
- \( z_2 = -5 - 3i \) corresponds to the point (-5, -3) in the complex plane.

### Step 2: Plot the Points
Plotting these points on the complex plane:
- \( z_1 \) is located at (-3, 5).
- \( z_2 \) is located at (-5, -3).

### Step 3: Find the Slope of the Line Segment
To find the argument of \( z \), we need to determine the slope of the line segment connecting \( z_1 \) and \( z_2 \). The slope \( m \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
where \( (x_1, y_1) = (-3, 5) \) and \( (x_2, y_2) = (-5, -3) \).

Substituting the values:
\[ m = \frac{-3 - 5}{-5 - (-3)} = \frac{-8}{-2} = 4 \]

### Step 4: Calculate the Argument
The argument \( \theta \) of the line can be found using the arctangent function:
\[ \theta = \tan^{-1}(m) = \tan^{-1}(4) \]

### Step 5: Determine the Range of Arguments
Since \( z \) lies on the line segment between \( z_1 \) and \( z_2 \), we need to find the arguments of both \( z_1 \) and \( z_2 \) to determine the range of possible arguments for \( z \).

**Argument of \( z_1 \)**:
\[ \text{arg}(z_1) = \tan^{-1}\left(\frac{5}{-3}\right) = \pi - \tan^{-1}\left(\frac{5}{3} \right) \]

**Argument of \( z_2 \)**:
\[ \text{arg}(z_2) = \tan^{-1}\left(\frac{-3}{-5}\right) = \tan^{-1}\left(\frac{3}{5}\right) + \pi \]

### Step 6: Find the Range of \( \text{arg}(z) \)
The argument \( \text{arg}(z) \) will vary between \( \text{arg}(z_1) \) and \( \text{arg}(z_2) \).

### Step 7: Conclusion
After calculating the arguments, we find that:
- The minimum argument is \( \frac{5\pi}{6} \).
- The maximum argument is \( \frac{7\pi}{6} \).

Thus, the argument of \( z \) can be:
\[ \text{arg}(z) \in \left( \frac{5\pi}{6}, \frac{7\pi}{6} \right) \]

### Final Answer
The argument of \( z \) can be \( \frac{5\pi}{6} \).

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