Question

Let $A(a, 0)$ , $B(0, b)$ and $C (1, 1)$ be three points. If $\frac{1}{a}+\frac{1}{b}=1$ , then the three points are

• A. vertices of an equilateral triangle
• B. vertices of a right angled triangle
• C. collinear
• D. vertices of an isosceles triangle

Answer 1

Answer: (C)

collinear

Answer 2

Area of $\Delta ABC = \begin{vmatrix}a&0&1\\\ 0&b&1\\\ 1&1&1\end{vmatrix}$
$= a\left(b-1\right) -0 + 1\left(0-b\right)$
$= ab - a -b = 0$
$\left[\because \frac{1}{a} +\frac{1}{b} = 1 \Rightarrow b + a = ab\right]$
$\therefore$ Points $A, B$ and $C$ are collinear.