If a complex number `z` satisfies `|2z+10+10 i|lt=5sqrt(3)-5,` then the least principal argument of `z` is

Options

A. `-(5pi)/6`
B. `-(11pi)/(12)`
C. `-(3pi)/4`
D. `-(2pi)/3`

Answer

To solve the problem, we start with the given inequality involving the complex number \( z \):

\[ |2z + 10 + 10i| \leq 5\sqrt{3} - 5 \]

### Step 1: Rewrite the inequality
We can rewrite the expression inside the modulus:

\[ |2z + 10 + 10i| = |2(z + 5 + 5i)| \]

This simplifies to:

\[ 2|z + 5 + 5i| \leq 5\sqrt{3} - 5 \]

### Step 2: Divide by 2
Next, we divide both sides of the inequality by 2:

\[ |z + 5 + 5i| \leq \frac{5\sqrt{3} - 5}{2} \]

### Step 3: Identify the center and radius
The expression \( |z + 5 + 5i| \leq r \) represents a circle in the complex plane, where the center is at \( -5 - 5i \) and the radius \( r \) is given by:

\[ r = \frac{5\sqrt{3} - 5}{2} \]

### Step 4: Calculate the radius
Now, we can calculate the radius:

\[ r = \frac{5(\sqrt{3} - 1)}{2} \]

### Step 5: Determine the position of the center
The center of the circle is at the point \( (-5, -5) \) in the complex plane, which is located in the third quadrant.

### Step 6: Find the least principal argument
To find the least principal argument of \( z \), we need to determine the angle from the origin (0,0) to the center of the circle (-5, -5). The angle \( \theta \) can be calculated using:

\[ \tan(\theta) = \frac{y}{x} = \frac{-5}{-5} = 1 \]

This gives us:

\[ \theta = \tan^{-1}(1) = \frac{\pi}{4} \]

Since the center is in the third quadrant, the actual angle is:

\[ \theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \]

### Step 7: Adjust for the radius
The radius \( r \) extends from the center towards the origin. The angle corresponding to the radius can be calculated as follows:

The angle from the center to the edge of the circle will be:

\[ \text{Least Principal Argument} = \frac{5\pi}{4} - \text{angle corresponding to radius} \]

Since the radius is less than the distance to the origin, we can find the least principal argument by considering the angle formed with the x-axis.

### Final Answer
Thus, the least principal argument of \( z \) is:

\[ \text{Least Principal Argument} = -\frac{5\pi}{6} \]