Find the range of `K` for which the equation `|z+i| - |z-i | = K` represents a hyperbola.

Options

A. `k in (-2,2)`
B. `k in [2,2]`
C. `k in (0,2)`
D. `k in (-2,0)`

Explanation

To solve the problem of finding the range of \( K \) for which the equation \( |z+i| - |z-i| = K \) represents a hyperbola, we can follow these steps:

### Step 1: Understand the Equation
The equation \( |z+i| - |z-i| = K \) describes the difference in distances from a point \( z \) in the complex plane to the points \( i \) (which is \( (0, 1) \) in Cartesian coordinates) and \( -i \) (which is \( (0, -1) \)).

### Step 2: Identify the Foci
The points \( i \) and \( -i \) are the foci of the hyperbola. The distance between these two points can be calculated as follows:
\[ \text{Distance} = |i - (-i)| = |i + i| = |2i| = 2 \]

### Step 3: Condition for Hyperbola
For the equation \( |z+i| - |z-i| = K \) to represent a hyperbola, the absolute value of \( K \) must be less than the distance between the foci. This is expressed mathematically as:
\[ |K| < \text{Distance between the foci} \]
Substituting the distance we found:
\[ |K| < 2 \]

### Step 4: Determine the Range of K
The inequality \( |K| < 2 \) implies:
\[ -2 < K < 2 \]
Thus, the range of \( K \) for which the equation represents a hyperbola is:
\[ K \in (-2, 2) \]

### Final Answer
The range of \( K \) for which the equation \( |z+i| - |z-i| = K \) represents a hyperbola is:
\[ K \in (-2, 2) \]

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