Prove that the points A(a, 0), B(0, b) and C(1, 1) are collinear if


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



Consider the points A(a, 0), B(0, b) and C(1, 1).
Here, (x1 = ay1 = 0), (x2 = 0, y2 = b) and (x3 = 1, y3 = 1).
It is given that the points are collinear. So,


$\Rightarrow a(b-1)+0(1-0)+1(0-b)=0$

$\Rightarrow a b-a-b=o$

Divid ing the equation by $a b$ :

$\Rightarrow 1-\frac{1}{b}-\frac{1}{a}=0$

$\Rightarrow 1-\left(\frac{1}{a}+\frac{1}{b}\right)=0$


Therefore, the given points are collinear if $\left(\frac{1}{a}+\frac{1}{b}\right)=1$.


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