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Prove that the Points (A, 0), (0, B) and (1, 1) Are Collinear If `1/A+1/B=1` - CBSE Class 10 - Mathematics

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Question

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

Solution

The formula for the area ‘A’ encompassed by three points(x1,y1),(x2,y2) and (x3,y3) is given by the formula,

We know area of triangle formed by three points (x1y1),(x2,y2) and (x3,y3)is given by 

Δ`=1/2[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]`

If three points are collinear the area encompassed by them is equal to 0.

The three given points are A(a,0), B(0,b) and C(1,1). 

A`=1/2[a(b-1)+1(0-b)]`

`=1/2[ab-a-b] ` 

It is given that `1/a+1/b=1`  

So we have, 

`1/a+1/b=1` 

`(a+b)/(ab)=1` 

a+b=ab 

Using this in the previously arrived equation for area we have, 

A`=ab-(a+b)`

A `ab-ab`

A=0 

Since the area enclosed by the three points is equal to 0, the three points need to be collinear

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Solution Prove that the Points (A, 0), (0, B) and (1, 1) Are Collinear If `1/A+1/B=1` Concept: Area of a Triangle.
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