Complex Numbers
Modulus and Argument
Grade Class 11
Question:
<p>If \( \arg\left(\dfrac{z-1}{z+1}\right) = \dfrac{\pi}{4} \), then the locus of \(z\) is:</p>
A circle of radius \sqrt{2}, centre (0,1)
A circle of radius 2, centre (1,0)
A parabola
A straight line
Step-by-Step Solution
Key Concept: arg((z-1)/(z+1)) = \pi/4 means the angle subtended by segment from -1 to 1 at point z is \pi/4 — this is an arc of a circle. The full circle has diameter from z=-1 to z=1 shifted to give radius \sqrt{2} and centre (0,1).
<p>The locus is an arc of the circle $x^2+(y-1)^2=2$, i.e., centre $(0,1)$, radius $\sqrt{2}$. (Upper semicircle gives arg=\pi/4 ✓.)</p>
Correct Answer: A