# If points (a, 0), (0, b) and (1, 1)

Question:

If points $(a, 0),(0, b)$ and $(1,1)$ are collinear, then $\frac{1}{a}+\frac{1}{b}=$

(a) 1

(b) 2

(c) 0

(d) −1

Solution:

We have three collinear points $\mathrm{A}(a, 0) ; \mathrm{B}(0, b) ; \mathrm{C}(1,1)$.

In general if $\mathrm{A}\left(x_{1}, y_{1}\right) ; \mathrm{B}\left(x_{2}, y_{2}\right) ; \mathrm{C}\left(x_{3}, y_{3}\right)$ are collinear then,

$x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$

So,

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

So,

$a b=a+b$

Divide both the sides by $(a b)$,

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