If P (z) is a variable point in the complex plane such that IM `(-1/z)=1/4` , then the value of the perimeter of the locus of P (z) is (use `pi=3.14` )

Answer

To solve the problem, we need to analyze the given condition and derive the equation of the locus of the point \( P(z) \) in the complex plane.

### Step-by-Step Solution:

1. **Assume \( z \) as a Complex Number**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

2. **Find \( -\frac{1}{z} \)**:
We need to compute \( -\frac{1}{z} \):
\[ -\frac{1}{z} = -\frac{1}{x + iy} \]
To simplify this, multiply the numerator and denominator by the conjugate of the denominator:
\[ -\frac{1}{z} = -\frac{x - iy}{x^2 + y^2} = \frac{-x + iy}{x^2 + y^2} \]

3. **Extract the Imaginary Part**:
The imaginary part of \( -\frac{1}{z} \) is:
\[ \text{IM}\left(-\frac{1}{z}\right) = \frac{y}{x^2 + y^2} \]
According to the problem, this is given to be \( \frac{1}{4} \):
\[ \frac{y}{x^2 + y^2} = \frac{1}{4} \]

4. **Cross-Multiply**:
Cross-multiplying gives:
\[ 4y = x^2 + y^2 \]

5. **Rearrange the Equation**:
Rearranging the equation leads to:
\[ x^2 + y^2 - 4y = 0 \]

6. **Complete the Square**:
To complete the square for the \( y \) terms:
\[ x^2 + (y^2 - 4y) = 0 \]
Completing the square for \( y^2 - 4y \):
\[ y^2 - 4y = (y - 2)^2 - 4 \]
Thus, the equation becomes:
\[ x^2 + (y - 2)^2 - 4 = 0 \]
Simplifying gives:
\[ x^2 + (y - 2)^2 = 4 \]

7. **Identify the Circle**:
The equation \( x^2 + (y - 2)^2 = 4 \) represents a circle with:
- Center at \( (0, 2) \)
- Radius \( r = 2 \)

8. **Calculate the Perimeter**:
The perimeter \( P \) of a circle is given by:
\[ P = 2\pi r \]
Substituting \( r = 2 \):
\[ P = 2 \cdot 3.14 \cdot 2 = 12.56 \]

### Final Answer:
The perimeter of the locus of \( P(z) \) is \( 12.56 \) units.

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