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Solve this following

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 :

  1. A hyperbola whose eccentricity is $\frac{2}{\sqrt{\mathrm{r}+1}}$,

    where $0

  2. An ellipse whose eccentricity is $\frac{1}{\sqrt{\mathrm{r}+1}}$,

    where $r>1$

  3. An ellipse whose eccentricity is $\frac{1}{\sqrt{\mathrm{r}+1}}$,

    where $r>1$

  4. An ellipse whose eccentricity is $\sqrt{\frac{2}{\mathrm{r}+1}}$,

    when $r>1$

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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