InterviewSolution
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InterviewSolution
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If three lines whose equations are `y=m_1x+c_1,y=m_2x+c_2`and `y=m_3x+c_3`are concurrent, then show that `m_1(c_2-c_3)+m_2(c_3-c_1)+m_3(c_1-c_2)=0`. |
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Answer» The given line are: `m_(1)x-y+c_(1)=0......(i)` `m_(2)x-y+c_(2)=0......(ii)` `m_(3)x-y+c_(3)=0......(iii)` On solving (i) and (ii) by cross multiplication, we get `(x)/((-c_(2)+c_(1)))=(y)/((m_(2)c_(1)-m_(1)c_(2)))=(1)/((-m_(1)+m_(2)))` `Rightarrow x=((c_(1)-c_(2)))/((m_(2)-m_(1)))and y=((m_(2)c_(1)-m_(1)c_(2)))/((m_(2)-m_(1)))` Thus, the point of intersection of (i) and (ii) is P `((c_(1)-c_(2))/(m_(2)-m_(1)),(m_(2)c_(1)-m_(1)c_(2))/(m_(2)-m_(1)))` Since the given three lines meet at a point, the point P must therefore lie on (ii) also. `therefore m_(3).((c_(1)-c_(2)))/((m_(2)-m_(1)))-((m_(2)c_(1)-m_(1)c_(2)))/((m_(2)-m_(1)))+c_(3)=0` `Rightarrow m_(3)(c_(1)-c_(2))-(m_(2)c_(1)-m_(1)c_(2))+c_(3)(m_(2)-m_(1))=0` `Rightarrow m_(1)(c_(2)-c_(3))-m_(2)(c_(3)-c_(1))-m_(3)(c_(1)-c_(2))=0` |
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