Chapter 8.4, Problem 63E

Solution

To find the slope of the asymptote in $\left(x,y\right)$ system.

The slope of the asymptotes in the $\left(x,y\right)$ system are $2-\sqrt{3}$ and $-2-\sqrt{3}$ .

Given information:

The slope of the asymptotes of a hyperbola in $\left(u,v\right)$ system is $\pm 1$ .

Calculation:

Since the slopes are $\pm 1$ , $\tan \theta =\pm 1$ . So, $\theta =\pm \frac{\pi }{4}$ . The angle formed by the slopes in $\left(u,v\right)$ system is $\frac{\pi }{4}$ .

The $\left(u,v\right)$ system is formed by rotating $\left(x,y\right)$ system by an angle $\frac{\pi }{6}$ . So, the slope of one asymptote must make angles $\frac{\pi }{4}-\frac{\pi }{6}=\frac{\pi }{12}$ .

The slopes will be $\tan \left(\frac{\pi }{12}\right)=2-\sqrt{3}$ . The other slope which is perpendicular to the first slope must be $-\frac{1}{2-\sqrt{3}}=-2-\sqrt{3}$ .

Conclusion:

The slope of the asymptotes in the $\left(x,y\right)$ system are $2-\sqrt{3}$ and $-2-\sqrt{3}$ .