Chapter A5.1, Problem 21E

To determine

To find: the hyperbola’s asymptotes and sketch the hyperbola.

Answer to Problem 21E

The hyperbola’s asymptote is $y=\pm x$

The sketch of the hyperbola is:

  

Explanation of Solution

Given:

  $x^2-y^2=1$

Calculation:

Given the equation of a hyperbola:

  $x^2-y^2=1$

Converting it to the standard form:

  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

  $\frac{x^2}{1^2}-\frac{y^2}{1^2}=1$

Therefore, $a=1,b=1$

Now

  $\begin{array}{l}c=\sqrt{a^2+b^2}\\ =\sqrt{1^2+1^2}\\ =\sqrt{2}\end{array}$

Foci are $\left(\pm c,0\right)$ , that is $\left(\pm \sqrt{2},0\right)$ .

Asymptotes are given by:

  $\begin{array}{l}y=\pm \frac{b}{a}x\\ \Rightarrow y=\pm x\end{array}$

Following is the sketch:

  

Conclusion:

The hyperbola’s asymptotes is:

  $y=\pm x$

The sketch of the hyperbola is: