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Question
Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axes respectively.
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Solution
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).
\[\therefore \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\]
\[ = \left| \frac{- \frac{4}{3} + 8}{1 + \frac{32}{3}} \right|\]
\[ \Rightarrow \tan \theta = \frac{4}{7}\]
Hence, the tangent of the angles between the lines is \[\frac{4}{7}\].
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