PROVING POINTS ARE COLLINEAR USING EQUATION OF THE LINE

Three or more points that lie on the same line are collinear points.

Step 1 :

Choose any two points and find equation of the line passes through those two points.

Step 2 :

Choose the remaining points and apply it in the equation.

Step 3 :

If the point satisfies the equation, then we can decide the given points are collinear.

If the point does not satisfy the equation, then we can decide the given points are non collinear.

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