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Question
Find the angles between the following pair of straight lines:
x − 4y = 3 and 6x − y = 11
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Solution
The equations of the lines are
x − 4y = 3 ... (1)
6x − y = 11 ... (2)
Let \[m_1 \text { and } m_2\] be the slopes of these lines.
\[m_1 = \frac{1}{4}, m_2 = 6\]
Let \[\theta\] be the angle between the lines.
Then,
\[\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\]
\[ = \left| \frac{\frac{1}{4} - 6}{1 + \frac{3}{2}} \right|\]
\[ = \frac{23}{10}\]
\[ \Rightarrow \theta = \tan^{- 1} \left( \frac{23}{10} \right)\]
Hence, the acute angle between the lines is \[\tan^{- 1} \left( \frac{23}{10} \right)\].
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