Question

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

Answer 1

1. If the question is like:

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

then the solution is:

Let A (a, 0), B (0, b) and C (1, 1) be the given points.

Suppose all given points are collinear.
Area of $\Delta$ABC = 0

$\Rightarrow \frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]=0\frac{1}{2}\left[a\right(b-1)+0(1-0)+1(0-b\left)\right]=0\frac{1}{2}[ab-a-b]=0ab-a-b=0$

Dividing both sides by ab, we get

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

Hence the given points are collinear only if when $\frac{1}{a}+\frac{1}{b}=1$