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Determinants

Question

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Find equation of line joining (1, 2) and (3, 6) using determinants.

Solution

Let P(x, y) be any point on the line joining A(1, 2) and B(3, 6).

therefore space space space area space of space triangle space ABP space equals space 0 therefore space space space space 1 half open vertical bar table row 1 cell space space space 2 end cell cell space space 1 end cell row 3 cell space space 6 end cell cell space space 1 end cell row straight x cell space space straight y end cell cell space space 1 end cell end table close vertical bar space equals space 0 space space space space space space space space rightwards double arrow space space space space space open vertical bar table row 1 cell space space space 2 end cell cell space space 1 end cell row 3 cell space space 6 end cell cell space space 1 end cell row straight x cell space straight y end cell cell space space 1 end cell end table close vertical bar space equals space 0 therefore space space space space space 1 open vertical bar table row 6 cell space space 1 end cell row straight y cell space space space 1 end cell end table close vertical bar space minus space 2 open vertical bar table row 3 cell space space 1 end cell row straight x cell space space 1 end cell end table close vertical bar plus 1 space open vertical bar table row 3 cell space space 6 end cell row straight x cell space space space straight y end cell end table close vertical bar space equals space 0 therefore space space space space 1 left parenthesis 6 minus straight y right parenthesis minus 2 left parenthesis 3 minus straight x right parenthesis plus 1 left parenthesis 3 straight y minus 6 straight x right parenthesis equals space 0 therefore space space space space space 6 minus straight y minus 6 plus 2 straight x plus 3 straight y minus 6 straight x space equals 0 therefore space space space space space minus 4 straight x plus 2 straight y space equals space 0 space space space space space space space space or space space 2 straight x minus straight y space equals space 0

which is required equation of line.

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