Question

If \( C \) is a point on the straight line joining the points \( A = -2+i \) and \( B = 3 - 4i \) in the Argand plane and } \[ \frac{AC}{CB} = \frac{1}{2}, \] then the argument of \( C \) is

• A. \( \tan^{-1}3 \)
• B. \( \tan^{-1}2 - \pi \)
• C. \( \tan^{-1}2 \)
• D. \( \pi - \tan^{-1}3 \)

Hint:

Use the section formula \( z = \frac{m z_2 + n z_1}{m+n} \) for internal division, and always adjust the argument based on the quadrant.

Answer 1

The Correct Option is B

Step 1: Use section formula in complex numbers.
Let \( A = -2+i \), \( B = 3-4i \). If \( C \) divides \( AB \) in the ratio \( 1:2 \), then \[ C = \frac{2A + 1B}{1+2} = \frac{2(-2+i) + (3 - 4i)}{3} = \frac{-4 + 2i + 3 - 4i}{3} = \frac{-1 - 2i}{3} = -\frac{1}{3} - \frac{2}{3}i \] Step 2: Find argument of \( C \)
\[ \arg(C) = \arg\left(-\frac{1}{3} - \frac{2}{3}i\right) = \tan^{-1}\left(\frac{2/3}{1/3}\right) - \pi = \tan^{-1}2 - \pi \]