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Question
Find the angles between the following pair of straight lines:
3x + y + 12 = 0 and x + 2y − 1 = 0
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Solution
The equations of the lines are
3x + y + 12 = 0 ... (1)
x + 2y − 1 = 0 ... (2)
Let
\[m_1 \text { and } m_2\] be the slopes of these lines.
\[m_1 = - 3, m_2 = - \frac{1}{2}\]
Let\[\theta\] be the angle between the lines.
Then,
\[\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\]
\[ = \left| \frac{- 3 + \frac{1}{2}}{1 + \frac{3}{2}} \right|\]
\[ = 1\]
\[ \Rightarrow \theta = \frac{\pi}{4}\text { or }45^\circ\]
Hence, the acute angle between the lines is \[ 45^\circ\]
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