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| Equation of the line passing through the intersection of two lines |
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| Let the lines be |
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 |
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| Let P (h,k) be the point of intersection of L1 = 0 and L2 = 0. |
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| Now consider the following equations, |
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 … (v) |
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 … (vi) |
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| This equation is a first degree in x and y and hence this represents a straight line. |
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| Again, from (iii) and (iv), the coordinates (h,k) satisfy (vi) for all the real values of l. |
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| (vi) represents lines through the point of intersection of lines L1 = 0 and L2 = 0. |
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Hence the set of lines passing through the intersection of lines  |
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| Remarks: |
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| The equation of a line contains two arbitrary constants. This means two geometrical conditions must be given to obtain the equation of a line. If a line passes through the intersection of two lines (given) only one geometrical condition is available. |
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| \ L1
+ lL2 = 0
contains one arbitrary constant l . A
second condition provided has to be used to determine the value of
l . |
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