Complex Numbers 2 Question 1
1. The equation $|z-i|=|z-1|, i=\sqrt{-1}$, represents
(a) a circle of radius $\frac{1}{2}$ (2019 Main, 12 April I)
(b) the line passing through the origin with slope 1
(c) a circle of radius 1
(d) the line passing through the origin with slope -1
Show Answer
Answer:
Correct Answer: 1. (b)
Solution:
- Let the complex number $z=x+i y$
Also given, $|z-i|=|z-1|$
$$ \begin{aligned} & \Rightarrow|x+i y-i|=|x+i y-1| \\ & \Rightarrow \sqrt{x^{2}+(y-1)^{2}}=\sqrt{(x-1)^{2}+y^{2}} \end{aligned} $$
$$ \left[\because|z|=\sqrt{(\operatorname{Re}(z))^{2}+(\operatorname{Im}(z))^{2}}\right] $$
On squaring both sides, we get
$x^{2}+y^{2}-2 y+1=x^{2}+y^{2}-2 x+1$
$\Rightarrow y=x$, which represents a line through the origin with slope 1.
