Question

Let $z\in C$ with $\operatorname{Im}\left( z \right)=10$ and it satisfies the equation $\dfrac{2z-n}{2z+n}=2i-1$ for some natural number$n$. Then \[\]
A.$n=20$ and $\operatorname{Re}(z)=-10$ \[\]
B. $n=20$ and $\operatorname{Re}(z)=10$\[\]
C. $n=40$ and $\operatorname{Re}(z)=-10$\[\]
D. $n=20$ and $\operatorname{Re}(z)=10$\[\]

Answer 1

Hint: Use the condition of equality between complex numbers where real and imaginary parts are compared.

Complete step by step answer:
 We know that the general form of a complex number is $z=a+ib$ where $a\in R$ is called the real part of $z$ and $b\in R$ is called the imaginary part of the complex number. The function $\operatorname{Re}\left( z \right)$ returns the real part of the complex number and the function $\operatorname{Im}\left( z \right)$ returns the imaginary part of the complex number. Two complex numbers are equal if and only if their respective real parts and imaginary parts are equal. In symbols
\[{{z}_{1}}={{z}_{2}}\Leftrightarrow \operatorname{Re}\left( {{z}_{1}} \right)=\operatorname{Re}\left( {{z}_{2}} \right)\text{ and Im}\left( {{z}_{1}} \right)=\operatorname{Im}\left( {{z}_{2}} \right)..(1)\]
The given equation involving a complex variable $z$ is ,
\[\dfrac{2z-n}{2z+n}=2\left( i-1 \right)\]
As given in the question $\operatorname{Im}\left( z \right)=10$. Let us take the real part of complex numbers $x$. Now the complex number is $z=x+10i$ .
\[\begin{align}
  & \dfrac{2z-n}{2z+n}=2\left( i-1 \right) \\
 & \Rightarrow \dfrac{2\left( x+10i \right)-n}{2\left( x+10i \right)+n}=2\left( i-1 \right) \\
 & \Rightarrow 2\left( x+10i \right)-n=2\left( i-1 \right)\left[ \left( 2x+n \right)+20i \right] \\
 & \Rightarrow \left( 2x-n \right)+20i=2\left( 20 \right)\left( -1 \right)-\left( 2x+n \right)+i\left[ 2\left( 2x+n \right)-20i \right] \\
\end{align}\]
We see that in the above result there is a complex number on both the left and right hand side. So let us use the formula (1) and compare real part of left hand side with real part of right hand side and imaginary part of left hand side with imaginary part of right hand side in the above equation. So
\[\begin{align}
  & 2x-n=-40-\left( 2x+n \right)\Rightarrow 4x=-40\Rightarrow x=-10 \\
 & 20=2\left( 2x+n \right)-20\Rightarrow 4x+2n=40\Rightarrow -40+2n=40\Rightarrow n=40 \\
\end{align}\]
So the real part of the complex number is $\operatorname{Re}\left( z \right)=-10$ and the value of $n$ is 40.

So, the correct answer is “Option C”.

Note: The question tests your knowledge of equality between complex numbers using \[\operatorname{Re}\left( z \right),Im\left( z \right)\] functions. Careful solution of simultaneous equation and use of formula will help you to arrive at the correct result. It is to be noted that the complex numbers are not in order which means only equality can be defined between complex numbers not inequality.