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Question
Find the acute angle between the lines 2x − y + 3 = 0 and x + y + 2 = 0.
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Solution
The equations of the lines are
2x − y + 3 = 0 ... (1)
x + y + 2 = 0 ... (2)
Let \[m_1 \text { and } m_2\] be the slopes of these lines.
\[m_1 = 2, m_2 = - 1\]
Let \[\theta\] be the angle between the lines.
Then,
\[\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\]
\[ = \left| \frac{2 + 1}{1 - 2} \right|\]
\[ = 3\]
\[ \Rightarrow \theta = \tan^{- 1} \left( 3 \right)\]
Hence, the acute angle between the lines is \[\tan^{- 1} \left( 3 \right)\].
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