Question

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

Answer 1

The formula for the area ‘A’ encompassed by three points, and is given by the formula,

We know area of triangle formed by three points is given by

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).

$$\begin{gathered} A=\frac{1}{2}\left|a(b-1)+1(0-b)\right| \\ =\frac{1}{2}\left|ab-a-b\right| \end{gathered}$$

It is given that

So we have,

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

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