The equation $\arg \left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}$ represents a circle with:

(B) centre at $(0,1)$ and radius $\sqrt{2}$

(A) centre at $(0,1)$ and radius 2

(C) centre at $(0,-1)$ and radius $\sqrt{2}$

(D) centre at $(0,0)$ and radius $\sqrt{2}$

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Complex Numbers

The equation

ar g (\frac{z - 1}{z + 1}) = \frac{π}{4}

represents a circle with:

Answer: (B)

With

z = x + i y

, multiplying numerator and denominator by the conjugate of the denominator gives

\frac{z - 1}{z + 1} = \frac{( x - 1 ) + i y}{( x + 1 ) + i y} = \frac{( x ^{2} + y ^{2} - 1 ) + 2 i y}{( x + 1 ) ^{2} + y ^{2}} .

An argument of

\frac{π}{4}

needs the real and imaginary parts to be equal and both positive, so

x^{2} + y^{2} - 1 = 2 y \Rightarrow x^{2} + (y - 1)^{2} = 2.

This is a circle with centre

(0, 1)

and radius

2

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Single correct option