Question

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

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

Answer 1

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

$\frac{y^2}{1+r}-\frac{x^2}{1-r}=1$
As $r>1$, $S$ is an ellipse
$\frac{y^2}{1+r}+\frac{x^2}{r-1}=1$
So, $a^2=1+r,b^2=r-1$
$\Rightarrow c^2=a^2-b^2=2$
$\therefore e=\frac{c}{a}=\sqrt{\frac{2}{r+1}}$
If $0<r<1$, $S$ is a hyperbola.
$\frac{y^2}{1+r}-\frac{x^2}{1-r}=1\Rightarrow a^2=1+r,b^2=1-r\Rightarrow e=\sqrt{1+\frac{b^2}{a^2}}\Rightarrow e=\sqrt{\frac{2}{1+r}}$