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

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

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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

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.5 | Q 23 | Page 54
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