Position of two points with respect to a line:

Let given line be ax + by + c = 0 and P(x₁, y₁), Q (x₂, y₂) be 2 given points. Now

 if  

=> both ax₁ + by₁ + c and ax₂ + by₂ + c are of opposite sign, then points P and  Q will be located on the opposite side of line ax + by + c = 0 and,

 if      

=> both ax₁ + by₁ + c and ax₂ + by₂ + c are of same sign, then the  points P and Q will lie on the same side of the given line. 

Illustration: Find range of a such that the point (a², 2a) and (2, 6) lies on same side of the line x + y -1 = 0.

Key concept: Sign of x + y -1 must be same for both the points.

Solution: For (2, 6) x + y - 1 = 2 + 6 -1 is  + ve. Thus a² + 2a -1 > 0

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