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Question
Write the condition of collinearity of points (x1, y1), (x2, y2) and (x3, y3).
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Solution
The condition for co linearity of three points` (x_1 , y_1) , (x_2 , y_2) " and " (x_3 , y_3 )` is that the area enclosed by them should be equal to 0.
The formula for the area ‘A’ encompassed by three points ` (x_1 , y_1) , (x_2 , y_2) " and " (x_3 , y_3 )` and is given by the formula,
`A = 1/2 |[x_1-x_2 y_1 - y_2],[x_2 - x_3 y_2 - y_3 ]|`
`A = 1/2 | (x_2 - x_2 )(y_2 - y_3 ) -(x_2 - x_3 )(y_1 - y_2 )|`
Thus for the three points to be collinear we need to have,
`1/2 | (x_1 - x_2 )( y_2 - y_3 )- (x_2 - x_3 )(y_1 -y_2)|=0`
`|(x_1 -x_2)(y_2 - y_3)-(x_2 - x_3)(y_1 - y_2)|=0`
The area ‘A’ encompassed by three points `(x_1 , y_1) , (x_2 , y_2) " and " (x_3 , y_3 )` , is also given by the formula,
`A = 1/2 |x_1 (y_2 - y_3 ) + x_2 (y_3 - y_1) + x_3 (y_1 - y_2 )|`
Thus for the three points to be collinear we can also have,
` 1/2 |x_1 (y_2 - y_3 ) + x_2 (y_3 - y_1) + x_3 (y_1 - y_2 )|`= 0
`x_1 (y_2 - y_3)+x_2 (y_3-y_1)+x_3(y_1 - y_2) = 0`
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