Question

If \(z_1 = 1 + i,\; z_2 = -2 + 3i,\; z_3 = \frac{ai}{3}\) are collinear, where \(i^2 = -1\), then value of \(a\) is:

• A. \(-1\)
• B. 3
• C. 4
• D. 5

Hint:

For collinearity, always set imaginary part of ratio = 0.

Answer 1

The Correct Option is D

Concept: Three complex numbers are collinear if: \[ \frac{z_3 - z_1}{z_2 - z_1} \in \mathbb{R} \]

Step 1:Compute differences \[ z_2 - z_1 = (-2+3i) - (1+i) = -3 + 2i \] \[ z_3 - z_1 = \frac{ai}{3} - (1+i) = -1 + i\left(\frac{a}{3} - 1\right) \]

Step 2:Form ratio \[ \frac{-1 + i\left(\frac{a}{3} - 1\right)}{-3 + 2i} \] Multiply by conjugate: \[ \frac{\left[-1 + i\left(\frac{a}{3} - 1\right)\right](-3 - 2i)}{13} \]

Step 3:Expand numerator Real part: \[ (-1)(-3) + \left(\frac{a}{3}-1\right)(-2) = 3 - \frac{2a}{3} + 2 = 5 - \frac{2a}{3} \] Imaginary part: \[ (-1)(-2) + \left(\frac{a}{3}-1\right)(-3) = 2 - a + 3 = 5 - a \]

Step 4:Condition for real Imaginary part = 0: \[ 5 - a = 0 \Rightarrow a = 5 \] Conclusion \[ {(D)\ 5} \]