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 solve the problem, we need to analyze the given equation:

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

This equation represents the difference in distances from the point \((x, y)\) to the foci of the hyperbola located at \((1, 0)\) and \((-1, 0)\).

### Step 1: Understanding the structure of the hyperbola

The standard form of a hyperbola is defined by the difference of distances from any point on the hyperbola to the two foci being a constant. For a hyperbola with foci at \((c, 0)\) and \((-c, 0)\), the relationship is given by:

\[ |S_1 - S_2| = 2a \]

where \(S_1\) and \(S_2\) are the distances to the foci, and \(2a\) is the constant difference.

### Step 2: Identifying the foci

In our case, the foci are at \((1, 0)\) and \((-1, 0)\). Thus, we can identify:

- \(S_1 = \sqrt{(x - 1)^2 + y^2}\)
- \(S_2 = \sqrt{(x + 1)^2 + y^2}\)

### Step 3: Setting up the equation

From the definition of the hyperbola, we have:

\[ |S_1 - S_2| = k \]

This means that:

\[ k = |S_1 - S_2| = | \sqrt{(x - 1)^2 + y^2} - \sqrt{(x + 1)^2 + y^2} | \]

### Step 4: Relating \(k\) to \(2a\)

For the hyperbola to exist, we need:

\[ k = 2a \]

where \(a\) is the semi-major axis.

### Step 5: Finding the value of \(a\)

To find the value of \(a\), we note that the distance between the foci is \(2c\), where \(c = 1\) (the distance from the origin to either focus). The relationship between \(a\), \(b\), and \(c\) in hyperbolas is given by:

\[ c^2 = a^2 + b^2 \]

Since \(c = 1\), we have:

\[ 1 = a^2 + b^2 \]

### Step 6: Determining the constraints on \(k\)

Since \(k = 2a\), we can express \(a\) in terms of \(k\):

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

Substituting this into the equation \(1 = a^2 + b^2\), we have:

\[ 1 = \left(\frac{k}{2}\right)^2 + b^2 \]

This implies:

\[ b^2 = 1 - \frac{k^2}{4} \]

For \(b^2\) to be non-negative:

\[ 1 - \frac{k^2}{4} \geq 0 \]

This leads to:

\[ \frac{k^2}{4} \leq 1 \]

or

\[ k^2 \leq 4 \]

Thus, we find:

\[ |k| \leq 2 \]

### Conclusion

Since \(k\) must be positive (as it represents a distance), we conclude:

\[ k < 2 \]

Thus, \(k\) belongs to the set of values:

\[ k \in (0, 2) \]