Question

The points represented by the complex numbers $1+ i , -2 + 3i , \frac{5}{3} i$ on the argand plane are

• A. vertices of an equilateral triangle
• B. vertices of an isosceles triangle
• C. collinear
• D. None of these

Answer 1

Answer: (C)

Let $z_1 = 1 + i, z_2 = - 2 + 3i$ and $z_3 = 0 + \frac{5}{3} i $
Then $\begin{vmatrix}x_{1}&y_{1}&1\\ x_{2}&y_{2}&1\\ x_{3}&y_{3}&1\end{vmatrix} = \begin{vmatrix}1&1&1\\ -2&3&1\\ 0&5/3&1\end{vmatrix} $
$ = 1\left(3- \frac{5}{3}\right) + 1\left(2\right) +1\left(\frac{-10}{3}\right) $
$= \frac{4}{3} + 2 - \frac{10}{3} $
$= \frac{4+6-10}{3} =0 $