Complex Numbers
Complex Plane / Geometry
Grade Class 11
Question:
<p>If \( z = x+iy \) and \( \arg\left(\dfrac{z-1}{z+1}\right) = \dfrac{\pi}{4} \), then the locus of \( z \) is:</p>
A straight line through origin
Arc of circle, centre (0,1)
Arc of circle, centre (0,-1)
Circle with centre at origin
Step-by-Step Solution
Key Concept: arg((z-1)/(z+1)) = \pi/4 represents a circular arc (locus of points from which the segment from -1 to 1 subtends angle \pi/4). The full circle passes through \pm1 with centre on the y-axis.
<p>The locus is an arc of a circle passing through $\pm 1$. The centre is on the perpendicular bisector (y-axis) at $(0,-1)$ (for arg = \pi/4, upper arc). Equation: $x^2+(y+1)^2 = 2$, upper semicircle.</p>
Correct Answer: C