PROVING POINTS ARE COLLINEAR USING EQUATION OF THE LINE

Example :

By using the concept of the equation of the straight line, prove that the given three points are collinear.

(i)  (4, 2) (7, 5) and (9, 7)

(ii)  (1,4) (3,-2) and (-3,16)

(i) Answer :

Equation of the line :

Let the given points be A (4, 2) B (7, 5) and C (9, 7).

Finding equation of the line passes through the two points A and B :

(y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)

(y - 2)/(5 - 2) = (x  - 4)/(7 - 4)

(y - 2)/3 = (x - 4)/3

(y - 2) = (x - 4)

x - y - 4 + 2 = 0

x - y - 2 = 0

By applying the remaining point C on the equation of the line AB.

9 - 7 - 2 = 0

9 - 9 = 0

0 = 0

Since the equation satisfies, the given points are collinear.

(ii) Answer :

Let the given points be A(1, 4), B(3, -2) and C(-3, 16)

Finding equation of the line passes through the two points A and B :

(y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)

(y - 4)/(-2 - 4) = (x - 1)/(3 - 1)

(y - 4)/(-6) = (x - 1)/2

2(y - 4) = -6(x - 1)

2y - 8 = -6x + 6

6x + 2y - 6 - 8 = 0

6x + 2y - 14 = 0

Divide the whole equation by 2 we get,

3x + y - 7 = 0

By applying the remaining point C on the equation of the line AB.

3(-3) + 16 - 7 = 0

-9 + 16 - 7 = 0

 -16 + 16 = 0

 0 = 0

Since the equation satisfies, the given points are collinear.

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