Question
The equation $$\frac{{{x^2}}}{{1 - r}} - \frac{{{y^2}}}{{1 + r}} = 1,\,\,\,r > 1$$ represents :
A.
an ellipse
B.
a hyperbola
C.
a circle
D.
none of these
Answer :
none of these
Solution :
Given that $$\frac{{{x^2}}}{{1 - r}} - \frac{{{y^2}}}{{1 + r}} = 1,\,\,\,r > 1$$
$$\eqalign{
& {\text{As }}r > 1 \cr
& \therefore 1 - r < 0{\text{ and }}1 + r > 0 \cr
& \therefore {\text{Let }}1 - r = - {a^2},\,\,\,1 + r = {b^2},\,\,{\text{then we get}} \cr
& \frac{{{x^2}}}{{ - {a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\,\,\,\,\,\,\,\,\, \Rightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = - 1 \cr} $$
which is not possible for any real values of $$x$$ and $$y.$$