Chapter 1.1, Problem 23P

Any curve with the property that whenever it intersects a curve of a given family it does so at an angle $a\ne \pi /2$ is called an oblique trajectory of the given family. (See Figure 1.1.7) Let $m_1$ (equal to $\tan a_1$ ) denote the slope of the required family at the point $\left(x,y\right)$ and let $m_2$ (equal to $\tan a_2$ ) denote the slope of the given family. Show that

$m_1=\frac{m_2-\tan a}{1+m_2\tan a}$.

[Hint: From Figure 1.1.7, $\tan a_1=\tan \left(a_2-a\right)$]. Thus, the equation of the family of oblique trajectories is obtained by solving

$\frac{dy}{dx}=\frac{m_2-\tan a}{1+m_2\tan a}$.

$\begin{array}{l}{m}_1=\tan a_1=\text{slope}\text{of}\text{required}\text{family}\\ m_2=\tan a_2=\text{slope}\text{of}\text{given}\text{family}\end{array}$

Figure 1.1.7: Oblique trajectories intersect at an angle a.