Question

Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axes respectively.

Answer 1

The respective equations of the lines having intercepts 3, 4 and 1, 8 on the axes are

$\frac{x}{3}+\frac{y}{4}=1$ ... (1)

$\frac{x}{1}+\frac{y}{8}=1$ ... (2)

Let m₁ and m₂ be the slope of the lines (1) and (2), respectively.

$\therefore m_1=-\frac{4}{3},m_2=-8$

Let $\theta$ be the angle between the lines (1) and (2).

$$\begin{gathered} \therefore \tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right| \\ =\left|\frac{{\displaystyle -\frac{4}{3}+8}}{1+{\displaystyle \frac{32}{3}}}\right| \\ \Rightarrow \tan \theta =\frac{{\displaystyle 4}}{{\displaystyle 7}} \end{gathered}$$

Hence, the tangent of the angles between the lines is $\frac{4}{7}$.