Let S=x,y∈R^2:y^21+r-x^21-r=1, where r≠±1. Then S represents:

Q47 / 90 · JEE Main 2019 (10 Jan Shift 2) · Mathematics

Let $S = \{(x, y) \in R^{2} : \frac{y^{2}}{1 + r} - \frac{x^{2}}{1 - r} = 1\},$ where $r \neq \pm 1 .$ Then $S$ represents:

A
An ellipse whose eccentricity is $\frac{1}{\sqrt{r + 1}},$ when $r > 1 .$
B
A hyperbola whose eccentricity is $\frac{2}{\sqrt{r + 1}},$ when $0 < r < 1 .$
Correct answer
C
An ellipse whose eccentricity is $\sqrt{\frac{2}{r + 1}},$ when $r > 1$
D
A hyperbola whose eccentricity is $\frac{2}{\sqrt{1 - r}},$ when $0 < r < 1$

Given

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

r \in (0, 1)

then this curve is a hyperbola whose eccentricity is

\sqrt{1 + \frac{1 - r}{1 + r}} = \sqrt{\frac{2}{r + 1}}

.

If

r > 1

, then this curve is an ellipse whose
eccentricity is

\sqrt{1 - \frac{r - 1}{r + 1}} = \sqrt{\frac{2}{r + 1}}

Hence, the correct answer is option: 'C'.