Collinearity of Points

Here you will learn condition for the collinearity of points with examples.

Let’s begin –

Collinearity of Points

If the given points lie in the same line, then the given points are collinear otherwise they are non-collinear.

Condition for collinearity of three given points

Three given points A(\(x_1,y_1\)), B(\(x_2,y_2\)) and C(\(x_3,y_3\)) are collinear if any one of the following conditions are satisfied :

(i)  Area of triangle ABC is zero. i.e.

\(1\over 2\) \(\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1  \end{vmatrix}\) = 0   or,  \(\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1  \end{vmatrix}\) = 0

or,  |[\(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\)]| = 0

(ii)  Slope of AB = Slope of BC = Slope of AC. i.e. \(y_2-y_1\over {x_2-x_1}\) = \(y_3-y_2\over {x_3-x_2}\) = \(y_3-y_1\over {x_3-x_1}\)

(iii) AB + BC = AC  or,  AC + BC = AB  or,  AC + AB = BC  (Use distance formula to calculate this)

(iv)  find the equation of line passing through 2 given points, if the third point satisfies the given equation of the line, then three points are collinear.

Example : Prove that the points (a, b + c), (b, c + a) and (c, a + b) are collinear.

Solution : Let A = \((x_1, y_1)\) = (a, b + c), B = \((x_2, y_2)\) = (b, c + a) and C = \((x_3, y_3)\) = (c, a + b) be three given points. Then,

|[\(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\)]| = 0

 = a{(c + a) – (a + b)} + b{(a + b) – (b + c)} + c{(b + c) – (c + a)}

= a(c – b) + b(a – c) + c(b – a) = 0

So, the area of triangle is zero.

Hence, the given points are collinear.

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