Prove that the points A(a, 0), B(0, b) and C(1, 1) are collinear if $(\frac{1}{a} + \frac{1}{b}) = 1 .$

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Solution

Consider the points A(a, 0), B(0, b) and C(1, 1).
Here, (x₁ = a, y₁ = 0), (x₂ = 0, y₂ = b) and (x₃ = 1, y₃ = 1).
It is given that the points are collinear. So,
$x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) = 0 \Rightarrow a (b - 1) + 0 (1 - 0) + 1 (0 - b) = 0 \Rightarrow a b - a - b = o Divid ing the equation by a b : \Rightarrow 1 - \frac{1}{b} - \frac{1}{a} = 0 \Rightarrow 1 - (\frac{1}{a} + \frac{1}{b}) = 0 \Rightarrow (\frac{1}{a} + \frac{1}{b}) = 1$
Therefore, the given points are collinear if $(\frac{1}{a} + \frac{1}{b})$ = 1.

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