Let `S={(x,y) in R^(2):(y^(2))/(1+r)-(x^(2))/(1-r)=1}`, where `r ne pm 1`. Then S represents:

Options

A. A hyperbolawhose eccentricity is `(2)/(sqrt(r+1))`, where `0 lt r lt 1`.
B. An ellipse whose eccentricity is `(1)/(sqrt(r+1)), " where" r gt 1`
C. A hyperbola whose eccentricity is `(2)/(sqrt(1-r)), " where " 0 lt r lt 1`.
D. An ellipse whose eccentricity is `sqrt((2)/(r+1)), " where " r gt 1`.

Answer

To solve the problem, we need to analyze the equation given in the set \( S \):

\[ \frac{y^2}{1 + r} - \frac{x^2}{1 - r} = 1 \]

where \( r \neq \pm 1 \).

### Step 1: Analyze the equation for \( r > 1 \)

1. **Rewrite the equation**: When \( r > 1 \), \( 1 - r < 0 \). We can rewrite the equation as:
\[ \frac{y^2}{1 + r} + \frac{x^2}{r - 1} = 1 \]
This is the standard form of the equation of an ellipse.

2. **Identify parameters**: Here, we can identify:
- \( a^2 = r - 1 \)
- \( b^2 = 1 + r \)

3. **Calculate eccentricity**: The eccentricity \( E \) of an ellipse is given by:
\[ E = \sqrt{1 - \frac{a^2}{b^2}} = \sqrt{1 - \frac{r - 1}{1 + r}} \]
Simplifying this gives:
\[ E = \sqrt{\frac{(1 + r) - (r - 1)}{1 + r}} = \sqrt{\frac{2}{1 + r}} \]

### Step 2: Analyze the equation for \( 0 < r < 1 \)

1. **Rewrite the equation**: When \( 0 < r < 1 \), \( 1 - r > 0 \). The equation can be rewritten as:
\[ \frac{y^2}{1 + r} - \frac{x^2}{1 - r} = 1 \]
This is the standard form of the equation of a hyperbola.

2. **Identify parameters**: Here, we can identify:
- \( a^2 = 1 - r \)
- \( b^2 = 1 + r \)

3. **Calculate eccentricity**: The eccentricity \( E \) of a hyperbola is given by:
\[ E = \sqrt{1 + \frac{a^2}{b^2}} = \sqrt{1 + \frac{1 - r}{1 + r}} \]
Simplifying this gives:
\[ E = \sqrt{\frac{(1 + r) + (1 - r)}{1 + r}} = \sqrt{\frac{2}{1 + r}} \]

### Conclusion

From the analysis, we conclude that:

- For \( r > 1 \), the set \( S \) represents an ellipse with eccentricity \( E = \sqrt{\frac{2}{1 + r}} \).
- For \( 0 < r < 1 \), the set \( S \) represents a hyperbola with eccentricity \( E = \sqrt{\frac{2}{1 + r}} \).

### Final Answer
Thus, the correct representation of \( S \) is:
- An ellipse for \( r > 1 \)
- A hyperbola for \( 0 < r < 1 \)