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 1

The correct option is D

None of these

Explanation for correct option:

Standarize the given equation.

The equation $\frac{x^2}{1-r}-\frac{y^2}{1+r}=1$ and $r>1$ is given.

Assume that, $1-r=-a^2$ and $1+r=b^2$.

Therefore, the given equation becomes:

$$\begin{gathered} \frac{x^2}{-a^2}-\frac{y^2}{b^2}=1 \\ \Rightarrow -\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)=1 \\ \Rightarrow \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)=-1 \end{gathered}$$

Which is not possible for any real values of $x$ and $y$.

Therefore, option $D$ is correct.