If three points (x1, y1), (x2, y2) and (x3, y3) lie on the same line, prove that .

Engineering

Mathematics

Introduction to Determinants

Question

If three points (x₁, y₁), (x₂, y₂) and (x₃, y₃) lie on the same line, prove that $\frac{y_{2} - y_{3}}{x_{2} x_{3}} + \frac{y_{3} - y_{1}}{x_{3} x_{1}} + \frac{y_{1} - y_{2}}{x_{1} x_{2}} = 0$ .

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(x₁, y₁), (x₂, y₂), (x₃, y₃) lie on same line

To prove

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

Let the line be y = mx + c

(x₁, y₁) is on the line

⇒ y = mx₁ + c

(x₂, y₂) is on the line

⇒ y₂ = mx₂ + c

(x₃, y₃) is on the line

⇒ y₃ = mx₃ + c

LHS

\Rightarrow \frac{y_{2} - y_{3}}{x_{2} x_{3}} + \frac{y_{3} - y_{1}}{x_{3} x_{1}} + \frac{y_{1} - y_{2}}{x_{1} x_{2}}

\Rightarrow \frac{({mx}_{2} + c - {mx}_{3} - c)}{x_{2} x_{3}} + \frac{({mx}_{3} + c - {mx}_{1} - c)}{x_{1} x_{3}} + \frac{({mx}_{1} + c - {mx}_{2} - c)}{(x_{1} x_{2})}

\Rightarrow m [\frac{(x_{2} - x_{3})}{x_{2} x_{3}} + \frac{(x_{3} - x_{1})}{x_{3} x_{2}} + \frac{(x_{1} - x_{2})}{x_{1} x_{2}}]

\Rightarrow m [\frac{(x_{1} x_{2} - x_{1} x_{3})}{x_{1} x_{2} x_{3}} + \frac{(x_{2} x_{3} - x_{2} x_{1})}{x_{1} x_{2} x_{3}} + \frac{(x_{1} x_{3} - x_{2} x_{3})}{x_{1} x_{2} x_{3}}]

\Rightarrow m (\frac{0}{x_{1} x_{2} x_{3}}) \Rightarrow 0 \Rightarrow RHS

⇒ LHS = RHS

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