We will learn how to find the position of a point relative to a line and also the condition for two points to lie on the same or opposite side of a given straight line.
Let the equation of the given line AB be ax + by + C = 0…………….(i) and let the coordinates of the two given points P (x\(_{1}\), y\(_{1}\)) and Q (x\(_{2}\), y\(_{2}\)).
I: When P and Q are on opposite sides:
Let us assumed that the points P and Q are on opposite sides of the straight line.
The coordinate of the point R which divides the line joining P and Q internally in the ratio m : n are
(\(\frac{mx_{2} + nx_{1}}{m + n}\), \(\frac{my_{2} + ny_{1}}{m + n}\))
Since the point R lies on ax + by + C = 0 hence we must have,
a ∙ \(\frac{mx_{2} + nx_{1}}{m + n}\) + b ∙ \(\frac{my_{2} + ny_{1}}{m + n}\) + c = 0
⇒ amx\(_{2}\) + anx\(_{1}\) + bmy\(_{2}\) + bny\(_{1}\) + cm + cn = 0
⇒ m(ax\(_{2}\) + by\(_{2}\) + c )=  n(ax\(_{1}\) + by\(_{1}\) + c)
⇒ \(\frac{m}{n} =  \frac{ax_{1} + by_{1} + c}{ax_{2} + by_{2} + c}\)………………(ii)
II: When P and Q are on same sides:
Let us assumed that the points P and Q are on same side of the straight line. Now join P and Q. Now assume that the straight line, (produced) intersects at R.
The coordinate of the point R which divides the line joining P and Q externally in the ratio m : n are
(\(\frac{mx_{2}  nx_{1}}{m  n}\), \(\frac{my_{2}  ny_{1}}{m  n}\))
Since the point R lies on ax + by + C = 0 hence we must have,
a ∙ \(\frac{mx_{2}  nx_{1}}{m  n}\) + b ∙ \(\frac{my_{2}  ny_{1}}{m  n}\) + c = 0
⇒ amx\(_{2}\)  anx\(_{1}\) + bmy\(_{2}\)  bny\(_{1}\) + cm  cn = 0
⇒ m(ax\(_{2}\) + by\(_{2}\) + c )= n(ax\(_{1}\) + by\(_{1}\) + c)
⇒ \(\frac{m}{n} = \frac{ax_{1} + by_{1} + c}{ax_{2} + by_{2} + c}\)………………(iii)
Clearly, \(\frac{m}{n}\) is positive; hence, the condition (ii) is satisfied if (ax\(_{1}\)+ by\(_{1}\) + c) and (ax\(_{2}\) + by\(_{2}\) + c) are of opposite signs. Therefore, the points P (x\(_{1}\), y\(_{1}\)) and Q (x\(_{2}\), y\(_{2}\)) will be on opposite sides of the straight line ax + by + C = 0 if(ax\(_{1}\)+ by\(_{1}\) + c) and (ax\(_{2}\) + by\(_{2}\) + c) are of opposite signs.
Again, the condition (iii) is satisfied if (ax\(_{1}\)+ by\(_{1}\) + c) and (ax\(_{2}\) + by\(_{2}\) + c) have the same signs. Therefore, the points P (x\(_{1}\), y\(_{1}\)) and Q (x\(_{2}\), y\(_{2}\)will be on the same side of the line ax + by + C = 0 if (ax\(_{1}\)+ by\(_{1}\) + c) and (ax\(_{2}\) + by\(_{2}\) + c) have the same signs.
Thus, the two points P (x\(_{1}\), y\(_{1}\)) and Q (x\(_{2}\), y\(_{2}\)) are on the same side or opposite sides of the straight line ax + by + c = 0, according as the quantities (ax\(_{1}\)+ by\(_{1}\) + c) and (ax\(_{2}\) + by\(_{2}\) + c) have the same or opposite signs.
Remarks: 1. Let ax + by + c = 0 be a given straight line and P (x\(_{1}\), y\(_{1}\)) be a given point. If ax\(_{1}\)+ by\(_{1}\) + c is positive, then the side of the straight line on which the point P lies is called the positive side of the line and the other side is called its negative side.
2. Since a ∙ 0 + b ∙ 0 + c = c, hence it is evident that the origin is on the positive side of the line ax + by + c = 0 when c is positive and the origin is on the negative side of the line when c is negative.
3. The origin and the point P (x\(_{1}\), y\(_{1}\)) are on the same side or opposite sides of the straight line ax + by + c = 0, according as c and (ax\(_{1}\)+ by\(_{1}\) + c) are of the same or opposite signs.
Solved examples to find the position of a point with respect to a given straight line:
1. Are the points (2, 3) and (4, 2) on the same or opposite sides of the line 3x  4y  7 = 0?
Solution:
Let Z = 3x  4y  7.
Now the value of Z at (2, 3) is
Z\(_{1}\) (let) =3 × (2)  4 × (3)  7
= 6 + 12  7
= 18  7
= 11, which is positive.
Again, the value of Z at (4, 2) is
Z\(_{2}\) (let) = 3 × (4)  4 × (2)  7
= 12  8  7
= 12  15
= 3, which is negative.
Since, z\(_{1}\) and z\(_{2}\), are of opposite signs, therefore the two points (2, 3) and (4, 2) are on the opposite sides of the given line 3x  4y  7 = 0.
2. Show that the points (3, 4) and (5, 6) lie on the same side of the straight line 5x  2y = 9.
Solution:
The given equation of the straight line is 5x  2y = 9.
⇒ 5x  2y  9 = 0 ……………………… (i)
Now find the value of 5x  2y  9 at (3, 4)
Putting x = 3 and y = 4 in the expression 5x  2y  9 we get,
5 × (3)  2 × (4)  9 = 15  8  9 = 15  17 = 2, which is negative.
Again, putting x = 5 and y = 6 in the expression 5x  2y  9 we get,
5 × (5)  2 × (6)  9 = 25 + 12  9 = 13  9 = 32, which is negative.
Thus, the value of the expression 5x  2y  9 at (2, 3) and (4, 2) are of same signs. Therefore, the given two points (3, 4) and (5, 6) lie on the same side of the line given straight line 5x  2y = 9.
● The Straight Line
11 and 12 Grade Math
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