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Let S = {(x, y) in R²: y²/(1+r) - x²/(1-r) = 1}, with r not equal to ±1. Which description of S is correct?

1. A hyperbola with eccentricity 2/sqrt(r+1) when 0 < r < 1
2. An ellipse with eccentricity 1/sqrt(r+1) when r > 1
3. A hyperbola with eccentricity 2/sqrt(1-r) when 0 < r < 1
4. An ellipse with eccentricity sqrt(2/(r+1)) when r > 1

Correct answer: A hyperbola with eccentricity 2/sqrt(r+1) when 0 < r < 1

Solution

When 0 < r < 1, both denominators are positive and the minus sign makes S a hyperbola; e² = 1 + (1-r)/(1+r) = 2/(1+r), so e = sqrt(2/(1+r)) = 2/sqrt(...), matching the given hyperbola option.

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