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Question

Prove that the points (a, 0), (0, b) and (1, 1) are collinear if , 1a+1b=1.

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Solution

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

Area of triangle = 12|x1(y2y3)+x2(y3y1)+x3(y1y2)|=0

x1(y2y3)+x2(y3y1)+x3(y1y2)=0

a(b1)+0(10)+1(0b)=0

a+b=ab

1a+1b=1


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