Question

Let $S=\left\{\right(x,y)\in R^3:\frac{y^2}{1+r}-\frac{x^2}{1-r}=1\}$, where $r\ne \pm 1$. Then $S$ represents:

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

Answer 1

The correct option is D An ellipse whose eccentricity is $\sqrt{\frac{2}{r+1}}$, when $r>1$

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

for $r>1$, $\frac{y^2}{1+r}+\frac{x^2}{r-1}=1$

$e=\sqrt{1-\left(\frac{r-1}{r+1}\right)}$

$=\sqrt{\frac{(r+1)-(r-1)}{(r+1)}}$

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