Question

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

• A. an ellipse
• B. a hyperbola
• C. a circle
• D. none of these

Answer 1

The correct option is D

none of these

Explanation for the correct option:

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

We have, $1-r<0$ and $1+r>1$ since $r>1$.

Let, $1-r=-a^2$ and $1+r=b^2$

$\begin{array}{cc}\Rightarrow & -\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\ \Rightarrow & \frac{x^2}{a^2}+\frac{y^2}{b^2}=-1\end{array}$

We know that a number squared is always positive and the sum and quotient of positive numbers are also posiitve. Thus the $\text{LHS}$ is always positive.
But, the $\text{RHS}$ is negative.

This is impossible.

Hence, option D is correct.