Position of Point with Respect to Line

The general equation or standard equation of a straight line is given by

ax  + by + c = 0\,\,\,\,{\text{ -  -  - }}\left( {\text{i}} \right)



position-point-wrt-line

Let A\left( {{x_1},{y_1}} \right) be a point not lying on the line (i). Now draw a perpendicular from A to the X-axis through the point P as shown in the given diagram. It is clear from the diagram that {x_1} is the abscissa of the point P. If y is the ordinate of P, then \left(  {{x_1},y} \right) are the coordinates of P. Since P lies on the line (i), so it must satisfy equation of the line, i.e.

\begin{gathered} a{x_1} + by + c = 0 \\ \Rightarrow by =  - a{x_1} - c \\ \Rightarrow y =  - \frac{{a{x_1} + c}}{b}\,\,\,\,{\text{  -  -   - }}\left( {{\text{ii}}} \right) \\ \end{gathered}


Next we consider the difference {y_1} - y, i.e.

\begin{gathered} {y_1} - y = {y_1} - \left( { - \frac{{a{x_1}  + c}}{b}} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {y_1} +  \frac{{a{x_1} + c}}{b} = \frac{{b{y_1} + a{x_1} + c}}{b} \\ \Rightarrow {y_1} - y = \frac{{a{x_1} +  b{y_1} + c}}{b}\,\,\,\,{\text{ -  -  - }}\left( {{\text{iii}}} \right) \\ \end{gathered}


(a) If the point A is above the line, then {y_1} - y > 0. From equation (iii), we note that {y_1} - y > 0 only if \frac{{a{x_1} + b{y_1} + c}}{b} > 0 . But \frac{{a{x_1} + b{y_1} + c}}{b} > 0 if a{x_1} + b{y_1} + c > 0 and b > 0 or \frac{{a{x_1} + b{y_1} + c}}{b} > 0 if a{x_1} + b{y_1} + c < 0 and b < 0.
We conclude that the point A\left( {{x_1},{y_1}} \right) is above the line ax + by + c = 0 if
(i) a{x_1} + b{y_1} + c > 0 and b > 0
(ii) a{x_1} + b{y_1} +  c < 0 and b < 0

(b) If the point A is below the line, then {y_1} - y < 0. From equation (iii), we note that {y_1} - y < 0 only if \frac{{a{x_1} + b{y_1} + c}}{b} < 0 . But \frac{{a{x_1} + b{y_1} + c}}{b} < 0 if a{x_1} + b{y_1} + c < 0 and b > 0 or \frac{{a{x_1} + b{y_1} + c}}{b} < 0 if a{x_1} + b{y_1} + c > 0 and b < 0.
We conclude that the point A\left( {{x_1},{y_1}} \right) is below the line ax + by + c = 0 if
(i) a{x_1} + b{y_1} + c < 0 and b > 0
(ii) a{x_1} + b{y_1} +  c > 0 and b < 0

NOTE: The point A\left( {{x_1},{y_1}} \right) will be on the line ax + by + c = 0 if a{x_1}  + b{y_1} + c = 0.

Example: Determine whether the point \left( {1,3}  \right) lies below or above the line 3x  - 2y + 1 = 0?
Comparing the given line 3x  - 2y + 1 = 0 with general equation of line ax + by + c = 0, here we have a = 3, b  =  - 2 < 0 and c = 1.
Since \left( {1,3}  \right) is the given point, so {x_1}  = 1,\,{y_1} = 3.
Now

a{x_1}  + b{y_1} + c = 3\left( 1 \right) + \left( { - 2} \right)\left( 3 \right) + 1  =  - 2 < 0


Since a{x_1} + b{y_1} +  c < 0 and b < 0, so given point line lies above the line.

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