Let complex number 'z' satisfy the inequality `2 le | x| le 4`. A point P is selected in this region at random. The probability that argument of P lies in the interval `[-pi/4,pi/4]` is `1/K`, then K =

Answer

To solve the problem, we need to analyze the given conditions and find the value of K. Let's break it down step by step.

### Step 1: Understand the given inequalities
We are given that the complex number \( z \) satisfies the inequality \( 2 \leq |z| \leq 4 \). This means that the modulus of \( z \) lies between 2 and 4.

### Step 2: Interpret the modulus condition
The modulus \( |z| \) can be expressed as:
\[ |z| = \sqrt{x^2 + y^2} \]
where \( z = x + iy \). Thus, the inequalities can be rewritten as:
\[ 2 \leq \sqrt{x^2 + y^2} \leq 4 \]
Squaring all parts of the inequality gives:
\[ 4 \leq x^2 + y^2 \leq 16 \]
This represents the area between two circles: one with radius 2 and another with radius 4.

### Step 3: Visualize the region
The region defined by \( 4 \leq x^2 + y^2 \leq 16 \) is the annular region (ring) between the circles of radius 2 and radius 4 centered at the origin.

### Step 4: Determine the angle condition
Next, we need to find the probability that the argument of point \( P \) (which is a random point in the annular region) lies in the interval \( [-\frac{\pi}{4}, \frac{\pi}{4}] \). The argument \( \theta \) of a point \( P \) in polar coordinates can be expressed as:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

### Step 5: Calculate the favorable area
The angle \( \theta \) ranges from \( -\frac{\pi}{4} \) to \( \frac{\pi}{4} \). This corresponds to a sector of the annular region. The angle span is:
\[ \frac{\pi}{4} - (-\frac{\pi}{4}) = \frac{\pi}{2} \]

### Step 6: Calculate the total area of the annular region
The area of the annular region can be calculated as the difference between the areas of the two circles:
\[ \text{Area} = \pi(4^2) - \pi(2^2) = 16\pi - 4\pi = 12\pi \]

### Step 7: Calculate the area of the favorable sector
The area of the sector corresponding to the angle \( \frac{\pi}{2} \) in the annular region can be calculated as:
\[ \text{Area of sector} = \frac{\text{Angle}}{2\pi} \times \text{Total Area of annular region} = \frac{\frac{\pi}{2}}{2\pi} \times 12\pi = \frac{1}{4} \times 12\pi = 3\pi \]

### Step 8: Calculate the probability
The probability \( P \) that the argument of point \( P \) lies in the interval \( [-\frac{\pi}{4}, \frac{\pi}{4}] \) is given by the ratio of the area of the favorable sector to the total area of the annular region:
\[ P = \frac{\text{Area of sector}}{\text{Total Area}} = \frac{3\pi}{12\pi} = \frac{1}{4} \]

### Step 9: Relate to the given probability
According to the problem, this probability can also be expressed as \( \frac{1}{K} \). Thus, we have:
\[ \frac{1}{K} = \frac{1}{4} \]
From this, we can conclude that:
\[ K = 4 \]

### Final Answer
Thus, the value of \( K \) is:
\[ \boxed{4} \]