Question

The complex number $z$ which satisfies the condition $\left|\frac{i+z}{i-z}\right|=1$ lies on:

• A. Circle $x^2+y^2=1$
• B. The x-axis
• C. The y-axis
• D. The line $x+y=1$

Answer 1

Answer: (B)

The x-axis

Let $z=x+iy$

$\left|\frac{i+z}{i-z}\right|=1$

$\Rightarrow |i+z|=|i-Z|$

$\Rightarrow |x+yi+i|=|i-x-iy|$

$\Rightarrow |x+i(y+1)|=|-x+i(1-y)|$

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

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

$\Rightarrow y+1=1-y$

$\Rightarrow 2y=0$

$\Rightarrow y=0$

It $x$ present the $x-$axis