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: {Find slopes between points}
Using slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Step 2: {Calculate 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: {Conclusion}
Since all slopes are the same, the points are collinear.

Answer 2

Step 1: Identify the points on the Argand plane
The complex numbers correspond to points:
\(A = (1, 1)\), \(B = (-2, 3)\), and \(C = (0, \frac{5}{3})\).

Step 2: Use the slope formula to check collinearity
Calculate slope of line \(AB\):
\[ m_{AB} = \frac{3 - 1}{-2 - 1} = \frac{2}{-3} = -\frac{2}{3} \]
Calculate slope of line \(BC\):
\[ m_{BC} = \frac{\frac{5}{3} - 3}{0 - (-2)} = \frac{\frac{5}{3} - \frac{9}{3}}{2} = \frac{-\frac{4}{3}}{2} = -\frac{2}{3} \]

Step 3: Conclusion
Since \(m_{AB} = m_{BC} = -\frac{2}{3}\), the points are collinear.

Final Answer: Collinear