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Question Number 46816 by peter frank last updated on 31/Oct/18

Answered by tanmay.chaudhury50@gmail.com last updated on 01/Nov/18

slope M_1 =(((a/m_(2 ) )−(a/m_1 ))/(a(m_2 −m_1 )))       M_1 =((a(m_1 −m_2 ))/(am_1 m_2 (m_2 −m_1 )))=((−1)/(m_1 m_2 ))    slope M_2 =(((a/m_4 )−(a/m_3 ))/(a(m_4 −m_3 )))=((−1)/(m_3 m_4 ))  M_1 ×M_2 =−1    (since line 1⊥ to second line)  M_1 ×M_2 =((−1)/(m_1 m_2 ))×((−1)/(m_3 m_4 ))=−1  hence m_1 m_2 m_3 m_4 =−1   proved

$${slope}\:{M}_{\mathrm{1}} =\frac{\frac{{a}}{{m}_{\mathrm{2}\:} }−\frac{{a}}{{m}_{\mathrm{1}} }}{{a}\left({m}_{\mathrm{2}} −{m}_{\mathrm{1}} \right)} \\ $$$$\:\:\:\:\:{M}_{\mathrm{1}} =\frac{{a}\left({m}_{\mathrm{1}} −{m}_{\mathrm{2}} \right)}{{am}_{\mathrm{1}} {m}_{\mathrm{2}} \left({m}_{\mathrm{2}} −{m}_{\mathrm{1}} \right)}=\frac{−\mathrm{1}}{{m}_{\mathrm{1}} {m}_{\mathrm{2}} } \\ $$$$ \\ $$$${slope}\:{M}_{\mathrm{2}} =\frac{\frac{{a}}{{m}_{\mathrm{4}} }−\frac{{a}}{{m}_{\mathrm{3}} }}{{a}\left({m}_{\mathrm{4}} −{m}_{\mathrm{3}} \right)}=\frac{−\mathrm{1}}{{m}_{\mathrm{3}} {m}_{\mathrm{4}} } \\ $$$${M}_{\mathrm{1}} ×{M}_{\mathrm{2}} =−\mathrm{1}\:\:\:\:\left({since}\:{line}\:\mathrm{1}\bot\:{to}\:{second}\:{line}\right) \\ $$$${M}_{\mathrm{1}} ×{M}_{\mathrm{2}} =\frac{−\mathrm{1}}{{m}_{\mathrm{1}} {m}_{\mathrm{2}} }×\frac{−\mathrm{1}}{{m}_{\mathrm{3}} {m}_{\mathrm{4}} }=−\mathrm{1} \\ $$$${hence}\:{m}_{\mathrm{1}} {m}_{\mathrm{2}} {m}_{\mathrm{3}} {m}_{\mathrm{4}} =−\mathrm{1}\:\:\:{proved} \\ $$

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