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 the above

Hint:

Collinear points lie on a straight line, which can be confirmed by checking if the slopes between consecutive points are equal.

Answer 1

The Correct Option is C

Step 1: {Determine the slopes connecting the points}
Employ the slope formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Step 2: {Compute the slopes}
\[m_{AB} = \frac{3 - 1}{-2 - 1} = -\frac{2}{3}\]\[m_{BC} = \frac{\frac{5}{3} - 3}{0 - (-2)} = -\frac{2}{3}\]Step 3: {Inference}
As all calculated slopes are identical, the points lie on the same line.