Question

What does the equation $\frac{x^2}{r-1}-\frac{y^2}{1+r}=1,r>1,$ represents

Answer 1

$S=\left\{\right(x,y)ϵR^2;\frac{y^2}{1+r}-\frac{x^2}{1-r}=1\}$

$\frac{y^2}{1+r}+\frac{x^2}{r-1}=1$ _______ (1) if $r-1>0,r+1>0\Rightarrow r>1$ and $r>-1$

$\therefore r>1$ then equation (1) represent

an ellipse with eccentricity $e_E=\sqrt{1-\left(\frac{r-1}{r+1}\right)}=\sqrt{\frac{r+1-r+1}{r+1}}=\sqrt{\frac{2}{r+1}}$

If $r\in (-1,1),$ then (1) represents hyperbola with eccentricity

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

$\therefore$ It is ellipse $r>1$