If the equation `|sqrt((x - 1)^(2) + y^(2) ) - sqrt((x + 1)^(2) + y^(2))| = k ` represents a hyperbola , then k belongs to the set

Options

A. `(1, infty)`
B. `(0, 1)`
C. `(0, 2)`
D. `(0, infty)`

Answer

To determine the set of values for \( k \) such that the equation

\[ |\sqrt{(x - 1)^{2} + y^{2}} - \sqrt{(x + 1)^{2} + y^{2}}| = k \]

represents a hyperbola, we can follow these steps:

### Step 1: Understand the Form of the Equation
The given equation is of the form

\[ |s_1 - s_2| = k \]

where \( s_1 = \sqrt{(x - 1)^{2} + y^{2}} \) and \( s_2 = \sqrt{(x + 1)^{2} + y^{2}} \). This represents the difference in distances from any point \( (x, y) \) to the two foci located at \( (1, 0) \) and \( (-1, 0) \).

### Step 2: Identify the Properties of Hyperbola
For a hyperbola, the property states that the absolute difference of the distances from any point on the hyperbola to the two foci is a constant, which we denote as \( 2a \). Thus, we can equate:

\[ |s_1 - s_2| = 2a \]

### Step 3: Relate \( k \) to \( a \)
From the equation \( |s_1 - s_2| = k \), we can set:

\[ k = 2a \]

### Step 4: Determine the Value of \( a \)
The distance between the foci is \( 2 \) (from \( -1 \) to \( 1 \)), so the value of \( a \) must be less than the distance between the foci for the hyperbola to exist. Therefore, we have:

\[ a < 1 \]

### Step 5: Substitute \( a \) in Terms of \( k \)
Since \( k = 2a \), we can substitute:

\[ a = \frac{k}{2} \]

Thus, the condition \( a < 1 \) translates to:

\[ \frac{k}{2} < 1 \]

### Step 6: Solve for \( k \)
Multiplying both sides by \( 2 \):

\[ k < 2 \]

### Step 7: Conclusion
Since \( k \) must be a positive value (as it represents a distance), we conclude that:

\[ 0 < k < 2 \]

Thus, \( k \) belongs to the set \( (0, 2) \).