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Collinearity Condition

Prove that th...

Question

Prove that the points (a, b) , $(a_{1}, b_{1})$ and $(a - a_{1}, b - b_{1})$ are collinear if $a b_{1} = a_{1} b$ .

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Solution

If 3 points are collinear, then the area of the triangle formed by them is zero.

Area of triangle = $\frac{1}{2} | (x_{1} y_{2} - y_{1} x_{2}) + (x_{2} y_{3} - y_{2} x_{3}) + (x_{3} y_{1} - y_{3} x_{1}) | = 0$

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

$\Rightarrow (a b_{1} - b a_{1}) + a_{1} (b - b_{1}) - b_{1} (a - a_{1}) + (a - a_{1}) b - (b - b_{1}) a = 0$

$\Rightarrow a b_{1} - b a_{1} + a_{1} b - a_{1} b_{1} - b_{1} a + b_{1} a_{1} + a b - a_{1} b - b a + b_{1} a = 0$

$\Rightarrow a b_{1} = a_{1} b$

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