Question

If $z$ is a complex number satisfying $\left| z-i\operatorname{Re}\left( z \right) \right|=\left| z-im\left( z \right) \right|$, then $z$ lies on:
1. $y=x$
2. $y=-x$
3. $y=x+1$
4. $y=-x+1$

Answer 1

Hint: Complex number is a number generally represented as $z=a+ib$, where a and b is real number represented on real axis whereas $i$ is an imaginary unit represented on imaginary axis whose value is $i=\sqrt{-1}$. Modulus of a complex number is length of line segment on real axis is argument of matrix denoted by argument $\left( z \right)$ calculated by trigonometric value. Argument of complex numbers is denoted by $\arg \left( z \right)=\theta ={{\tan }^{-1}}\dfrac{b}{a}$.

Complete step by step answer:
According to the question it is asked to us to determine the value of $y$ if $z$ is a complex number and $z$ is satisfying $\left| z-i\operatorname{Re}\left( z \right) \right|=\left| z-im\left( z \right) \right|$. As we know that we represent a complex number by the summation or by subtraction of a real number and an imaginary number. So, we can write that a complex number is represented as $z=x+iy$, where both x and y are the real numbers but the iota makes it imaginary. So, as it is given to us that,
$\left| z-i\operatorname{Re}\left( z \right) \right|=\left| z-im\left( z \right) \right|$
And $z=x+iy$, so it can be written as:
$\begin{align}
  & \Rightarrow \left| x+iy-ix \right|=\left| x+iy-y \right| \\
 & \Rightarrow \left| x+i\left( y-x \right) \right|=\left| x-y+iy \right| \\
\end{align}$
If we solve this, then we get,
$\begin{align}
  & \Rightarrow {{x}^{2}}+{{\left( y-x \right)}^{2}}={{\left( x-y \right)}^{2}}+{{y}^{2}} \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}-2xy+{{x}^{2}}={{x}^{2}}-2xy+{{y}^{2}}+{{y}^{2}} \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}={{y}^{2}}+{{y}^{2}} \\
 & \Rightarrow {{x}^{2}}={{y}^{2}} \\
\end{align}$
And it can be written as $x=y$.

So, the correct answer is “Option 1”.

Note: Complex numbers are always written in the form of $z=a+ib$ where a and b are real numbers whereas $i$ is the imaginary part. We can convert a degree into radian by multiplying it by $\dfrac{\pi }{180}$. Argument of complex numbers is denoted by $\arg \left( z \right)=\theta ={{\tan }^{-1}}\dfrac{b}{a}$.