If the points (x1, y1), (x2, y2) and (x3, y3) are collinear, show that , i.e

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Locus of a Point

Various Form of a Straight Line

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Question

If the points (x₁, y₁), (x₂, y₂) and (x₃, y₃) are collinear, show that $\sum (\frac{y_{1} - y_{2}}{x_{1} x_{2}}) = 0$ , i.e $\frac{y_{1} - y_{2}}{x_{1} x_{2}} + \frac{y_{2} - y_{3}}{x_{2} x_{3}} + \frac{y_{3} - y_{1}}{x_{3} x_{1}} = 0$

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Given (x₁, y₁)(x₂, y₂)(x₃, y₃) are collinear

∣ \begin{array}{ccc} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array} ∣ = 0

x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂) = 0 ____ (1)

By solving first column

Divide (1) by x₁x₂x₃

\frac{x_{1} (y_{2} - y_{3})}{x_{1} x_{2} x_{3}} + \frac{x_{2} (y_{3} - y_{1})}{x_{1} x_{2} x_{3}} + \frac{x_{3} (y_{1} - y_{2})}{x_{1} x_{2} x_{3}} = 0

\frac{y_{2} - y_{3}}{x_{2} x_{3}} + \frac{y_{3} - y_{1}}{x_{1} x_{3}} + \frac{y_{1} - y_{2}}{x_{1} x_{2}} = 0

∴ \sum (\frac{y_{1} - y_{2}}{x_{1} x_{2}}) = 0

∴ Hence proved.