Question

If \[ \arg\left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}, \] then the locus of the point \(P(z)\) on the Argand plane is a

• A. line
• B. circle
• C. parabola
• D. hyperbola

Hint:

Remember the standard result: \[ \arg\left(\frac{z-z_1}{z-z_2}\right)=\alpha \] represents the locus of points from which the segment joining \(z_1\) and \(z_2\) is seen under a constant angle \(\alpha\). Therefore, the locus is a circle.

Answer 1

The Correct Option is B

Concept: For complex numbers, \[ \arg\left(\frac{z-z_1}{z-z_2}\right) \] represents the angle subtended by the line segment joining \(z_1\) and \(z_2\) at the point \(z\). If this angle is constant, the locus is a circle passing through \(z_1\) and \(z_2\).

Step 1: Identify the fixed points. Given \[ \arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{4} \] which can be written as \[ \arg\left(\frac{z-1}{z-(-1)}\right) = \frac{\pi}{4} \] The fixed points are \[ A(1,0) \] and \[ B(-1,0). \]

Step 2: Interpret geometrically. The condition \[ \arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{4} \] means \[ \angle APB = \frac{\pi}{4} \] where \(P(z)\) is the moving point.

Step 3: Use the constant angle theorem. The locus of a point that subtends a constant angle at a fixed chord \(AB\) is an arc of a circle through \(A\) and \(B\). Therefore the complete locus is a circle. \[\begin{aligned} \boxed{\text{Circle}} \end{aligned}\] Hence, option \(\mathbf{(B)}\) is correct.