If $\mathrm{C}$ is a point on the straight line joining the points $\mathrm{A}(-2+\mathrm{i})$ and…
If $\mathrm{C}$ is a point on the straight line joining the points $\mathrm{A}(-2+\mathrm{i})$ and $\mathrm{B}(3-4 \mathrm{i})$ in the Argand plane and $\frac{\mathrm{AC}}{\mathrm{CB}}=\frac{1}{2}$, then the argument of $\mathrm{C}$ is
$\tan ^{-1} 3$
$\tan ^{-1} 2-\pi$
$\tan ^{-1} 2$
$\pi-\tan ^{-1} 3$
Solution
Given the point $\mathrm{A}(-2,1), \mathrm{B}(3,-4)$
$\& \frac{\mathrm{AC}}{\mathrm{BC}}=\frac{1}{2}$
So, $\mathrm{C} \equiv\left(\frac{1 \times 3-4}{3}, \frac{-4+2}{3}\right)=\left(\frac{-1}{3}, \frac{-2}{3}\right)$
Now, argument of $C=\tan ^{-1}\left(\frac{\frac{-2}{3}}{\frac{-1}{3}}\right)-\pi$ $=\tan ^{-1}(2)-\pi$