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Question

Mathematics Question on complex numbers

The complex number zz satisfying the condition arg z−1z+1=π4\frac {z-1} {z+1} = \frac {\pi} {4}

A

a straight line

B

a circle

C

a parabola

D

none of these

Answer

a circle

Explanation

Solution

Let z=x+iyz = x + iy, then z−1z+1=x+iy−1x+iy+1\frac{z-1}{z+1}=\frac{x+iy-1}{x+iy+1} =(x−1)+iy(x+1)+iy⋅(x+1)−iy(x+1)−iy=\frac{\left(x-1\right)+iy}{\left(x+1\right)+iy} \cdot \frac{\left(x+1\right)-iy}{\left(x+1\right)-iy} =(x2+y2−1)+i(2y)(x+1)2+y2=\frac{\left(x^{2}+y^{2}-1\right)+i\left(2y\right)}{\left(x+1\right)^{2}+y^{2}} Since arg.z−1z+1=π4arg. \frac{z-1}{z+1}=\frac{\pi}{4} ∴tan π4=2 yx2+y2−1\therefore tan \,\frac{\pi}{4}=\frac{2\,y}{x^{2}+y^{2}-1} ⇒1=2yx2+y2−1\Rightarrow 1=\frac{2y}{x^{2}+y^{2}-1} ⇒x2+y2−1=2y\Rightarrow x^{2}+y^{2}-1=2y ⇒x2+y2−2y−1=0\Rightarrow x^{2}+y^{2}-2y-1=0 which represents a circle.