Question:
Let $S=\left\{(x, y) \in R^{2}: \frac{y^{2}}{1+r}-\frac{x^{2}}{1-r}=1\right\}, \quad$ where
$r \neq \pm 1$. Then $\mathrm{S}$ represents :
Correct Option: , 4
Solution:
$\frac{\mathrm{y}^{2}}{1+\mathrm{r}}-\frac{\mathrm{x}^{2}}{1-\mathrm{r}}=1$
for $r>1, \quad \frac{y^{2}}{1+r}+\frac{x^{2}}{r-1}=1$
$e=\sqrt{1-\left(\frac{r-1}{r+1}\right)}$
$=\sqrt{\frac{(r+1)-(r-1)}{(r+1)}}$
$=\sqrt{\frac{2}{r+1}}=\sqrt{\frac{2}{r+1}}$
Ontion (4)
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