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Question
Find the point of intersection of the following pairs of lines:
\[y = m_1 x + \frac{a}{m_1} \text { and }y = m_2 x + \frac{a}{m_2} .\]
Solution
The equations of the lines are \[y = m_1 x + \frac{a}{m_1} \text { and } y = m_2 x + \frac{a}{m_2} .\]
Thus, we have:
\[m_1 x - y + \frac{a}{m_1} = 0\] ... (1)
\[m_2 x - y + \frac{a}{m_2} = 0\] ... (2)
Solving (1) and (2) using cross-multiplication method:
\[\frac{x}{- \frac{a}{m_2} + \frac{a}{m_1}} = \frac{y}{\frac{a m_2}{m_1} - \frac{a m_1}{m_2}} = \frac{1}{- m_1 + m_2}\]
\[ \Rightarrow x = \frac{\frac{- a}{m_2} + \frac{a}{m_1}}{- m_1 + m_2}, y = \frac{\frac{a m_2}{m_1} - \frac{a m_1}{m_2}}{- m_1 + m_2}\]
\[ \Rightarrow x = \frac{a}{m_1 m_2} \text { and }y = \frac{a\left( m_1 + m_2 \right)}{m_1 m_2}\]
Hence, the point of intersection is \[\left( \frac{a}{m_1 m_2}, \frac{a\left( m_1 + m_2 \right)}{m_1 m_2} \right) \text { or }\left( \frac{a}{m_1 m_2}, a\left( \frac{1}{m^1} + \frac{1}{m_2} \right) \right)\].
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