Question:
Let
$\mathrm{S}=\left\{(x, y) \in \mathbf{R}^{2}: \frac{y^{2}}{1+\mathrm{r}}-\frac{x^{2}}{1-\mathrm{r}}=1\right\}$
where $r \neq \pm 1$ Then $S$ represents:
Correct Option: , 2
Solution:
Since, $r \neq \pm 1$, then there are two cases, when $r>1$
$\frac{x^{2}}{r-1}+\frac{y^{2}}{r+1}=1$ (Ellipse)
Then,
$(r-1)=(r+1)\left(1-e^{2}\right) \Rightarrow 1-e^{2}=\frac{(r-1)}{(r+1)}$
$\Rightarrow \quad e^{2}=1-\frac{(r-1)}{(r+1)}=\frac{2}{(r+1)}$
$\Rightarrow \quad e=\sqrt{\frac{2}{(r+1)}}$
When $0 $\frac{x^{2}}{1-r}-\frac{y^{2}}{1+r}=-1$ (Hyperbola) Then, $(1-r)=(1+r)\left(e^{2}-1\right) \Rightarrow e^{2}=1+\frac{(r-1)}{(r+1)}=\frac{2 r}{(r+1)}$ $\Rightarrow \quad e=\sqrt{\frac{2 r}{r+1}}$