The complex number having least positive argument and satisying `|z-5i| le 3`, is

Options

A. `12+16i`
B. `12/5 + (16i)/5`
C. `16/5+(12i)/5`
D. `-12/5+(16i)/5`

Explanation

To solve the problem of finding the complex number \( z \) that has the least positive argument and satisfies the condition \( |z - 5i| \leq 3 \), we will follow these steps:

### Step 1: Understand the Condition
The condition \( |z - 5i| \leq 3 \) represents a circle in the complex plane. The center of the circle is at the point \( 0 + 5i \) (which corresponds to the coordinates \( (0, 5) \)) and the radius is \( 3 \).

**Hint:** Remember that the expression \( |z - a| \) represents the distance from the complex number \( z \) to the point \( a \) in the complex plane.

### Step 2: Identify the Circle
The circle has:
- Center \( C(0, 5) \)
- Radius \( r = 3 \)

This means the circle will extend from \( (0, 5 - 3) = (0, 2) \) to \( (0, 5 + 3) = (0, 8) \) along the imaginary axis.

**Hint:** Sketching the circle can help visualize the problem and understand where the points lie.

### Step 3: Find the Points on the Circle
To find the points on the circle that have the least positive argument, we need to consider the tangent line to the circle that intersects the real axis. The tangent will be horizontal at the point where the circle is closest to the real axis.

**Hint:** The least positive argument corresponds to the point on the circle that is closest to the positive real axis.

### Step 4: Calculate the Coordinates of the Tangent Point
The distance from the center \( C(0, 5) \) to the real axis is \( 5 \). The radius is \( 3 \), so the closest point on the circle to the real axis will be at:
- \( y = 5 - 3 = 2 \)

This point is directly below the center along the imaginary axis, so the coordinates of this point are \( (0, 2) \).

**Hint:** The argument of a point in the complex plane is given by the angle it makes with the positive real axis.

### Step 5: Find the Argument
The point \( (0, 2) \) corresponds to the complex number \( 2i \). The argument of \( 2i \) is \( \frac{\pi}{2} \) (90 degrees), which is the least positive argument.

**Hint:** The argument can be calculated using the formula \( \tan^{-1}(\frac{y}{x}) \), but for points on the imaginary axis, it is straightforward.

### Step 6: Conclusion
The complex number that satisfies the condition \( |z - 5i| \leq 3 \) and has the least positive argument is:
\[ z = 2i \]

### Final Answer
Thus, the complex number is:
\[ z = 0 + 2i \]