Question

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

Answer 1

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

Area of triangle = $\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|=0$

$\Rightarrow x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)=0$

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

$\Rightarrow a+b=ab$

$\Rightarrow \frac{1}{a}+\frac{1}{b}=1$