Question

Find the angles between the following pair of straight lines:

(m² − mn) y = (mn + n²) x + n³ and (mn + m²) y = (mn − n²) x + m³.

Answer 1

The equations of the lines are
(m² − mn) y = (mn + n²) x + n³          ... (1)
(mn + m²) y = (mn − n²) x + m³         ... (2)
Let \[m_1 \text { and } m_2\]  be the slopes of these lines. 

\[\therefore m_1 = \frac{mn + n^2}{m^2 - mn}, m_2 = \frac{mn - n^2}{mn + m^2}\]

Let \[\theta\] be the angle between the lines.
Then,

\[\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\]

\[ = \left| \frac{\frac{mn + n^2}{m^2 - mn} - \frac{mn - n^2}{mn + m^2}}{1 + \frac{mn + n^2}{m^2 - mn} \times \frac{mn - n^2}{mn + m^2}} \right|\]

\[ \Rightarrow \tan \theta = \left| \frac{\left( mn + n^2 \right)\left( mn + m^2 \right) - \left( mn - n^2 \right)\left( m^2 - mn \right)}{\left( m^2 - mn \right)\left( mn + m^2 \right) + \left( mn + n^2 \right)\left( mn - n^2 \right)} \right|\]

\[ \Rightarrow \tan \theta = \left| \frac{m^2 n^2 + m^3 n + m n^3 + m^2 n^2 - m^3 n + m^2 n^2 + m^2 n^2 - m n^3}{m^3 n + m^4 - m^2 n^2 - m^3 n + m^2 n^2 - m n^3 + m n^3 - n^4} \right|\] 

\[\Rightarrow \tan\theta = \left| \frac{4 m^2 n^2}{m^4 - n^4} \right|\]

Hence, the acute angle between the lines is \[\tan^{- 1} \left( \frac{4 m^2 n^2}{m^4 - n^4} \right)\].