Chapter 10.4, Problem 64ES

Given two intersecting lines, let $L_2$ be the line with the larger angle of inclination ${\varphi }_2,$ and let $L_1$ be the line with the smaller angle of inclination ${\varphi }_1.$ We define the angle $\theta$ between $L_1$ and $L_2$ by $\theta ={\varphi }_2-{\varphi }_1.$ (See the accompanying figure.)

(a) Prove: If $L_1\text{and}{L}_2$ are not perpendicular, then $\tan \theta =\frac{m_2-m_1}{1+m_1{m}_2}$ where $L_1\text{and}{L}_2$ have slopes $m_1\text{and }{m}_2.$

(b) Prove Theorem 10.4.5. [Hint: Introduce coordinates so that the equation $\left(x^2/a^2\right)+\left(y^2/b^2\right)=1$ describes the ellipse, and use part (a).]

(c) Prove Theorem 10.4.6. [Hint: Introduce coordinates so that the equation $\left(x^2/a^2\right)-\left(y^2/b^2\right)=1$ describes the hyperbola, and use part (a).]