Complex Numbers
Complex Plane / Geometry
Grade Class 11
Question:
<p>On the complex plane, the locus of \( z \) satisfying \( \arg(z - 3) = \dfrac{\pi}{3} \) is:</p>
A ray from origin at angle \pi/3
A ray from (3,0) at angle \pi/3
A half-line from (3,0) making angle \pi/3 with positive x-axis
A straight line
Step-by-Step Solution
Key Concept: arg(z-3) = \pi/3 means the vector from the fixed point 3 to z makes angle \pi/3 with the positive real axis — this is a ray (half-line) starting at z = 3.
<p>$ \arg(z-3) = \pi/3 $ means $z-3$ has argument $\pi/3$, so $z = 3 + r e^{i\pi/3}$ for $r > 0$. This is a ray starting at $(3,0)$ in direction $(\cos\pi/3, \sin\pi/3)$.</p>
Correct Answer: C