Position of a point with respect to a given line:

Let given line be ax + by + c = 0, and we have to check whether the point (a, b) lies above the line or below the line. To check 1st draw a perpendicular line from P(a, b) on x-axis which intersect the given line at Q. x-coordinate of Q will be a and corresponding to a, y-coordinate of Q will be

. Now

if β>β', the point will be located above the line

    if β=β' point will be located on the line                                                         

and if β < β' point will be located below the line.

Illustration: If point (α, α²) lies between the acute angle region formed between lines y = 2x and y = 3x, then find the range of a.

Key concept : Since y co-ordinate of (α, α²) is always on non negative, point (α, α²) will always lie above the x axis. Now for the acute angle region the point (a, b) should lie below the line y = 3x and above the line y = 2x.      

Solution:  (α, β) should lie below the lie y = 3x  => 3α < α² and

                 (α, β) should also lie above the line y = 2x => α² > 2α

                 => 2α < α² < 3α  => 2 < α < 3  

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