If `|z – 2| ge |z – 4|` then correct statement is-

Options

A. `R(z)ge3`
B. `R(z)le3`
C. `R(z)ge2`
D. `R(z)le2`

Answer

To solve the inequality \( |z - 2| \geq |z - 4| \), we will follow these steps:

### Step 1: Define \( z \)
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number \( z \).

### Step 2: Rewrite the inequality
The inequality can be rewritten as:
\[ |z - 2| \geq |z - 4| \]
This translates to:
\[ | (x + iy) - 2 | \geq | (x + iy) - 4 | \]
Which simplifies to:
\[ | (x - 2) + iy | \geq | (x - 4) + iy | \]

### Step 3: Apply the modulus formula
Using the modulus formula \( |a + bi| = \sqrt{a^2 + b^2} \), we can express the inequality as:
\[ \sqrt{(x - 2)^2 + y^2} \geq \sqrt{(x - 4)^2 + y^2} \]

### Step 4: Square both sides
To eliminate the square roots, we square both sides:
\[ (x - 2)^2 + y^2 \geq (x - 4)^2 + y^2 \]

### Step 5: Simplify the equation
Now, we can simplify the equation by canceling \( y^2 \) from both sides:
\[ (x - 2)^2 \geq (x - 4)^2 \]

### Step 6: Expand both sides
Expanding both sides gives us:
\[ x^2 - 4x + 4 \geq x^2 - 8x + 16 \]

### Step 7: Cancel \( x^2 \)
We can cancel \( x^2 \) from both sides:
\[ -4x + 4 \geq -8x + 16 \]

### Step 8: Rearrange the inequality
Rearranging the terms results in:
\[ -4x + 8x \geq 16 - 4 \]
\[ 4x \geq 12 \]

### Step 9: Solve for \( x \)
Dividing both sides by 4 gives:
\[ x \geq 3 \]

### Conclusion
Since \( x \) is the real part of \( z \), we conclude that:
\[ \text{Re}(z) \geq 3 \]