# Let z

Question:

Let $z \in \mathrm{C}$ with $\operatorname{Im}(z)=10$ and it satisfies $\frac{2 z-n}{2 z+n}=2 i-1$ for some natural number $n$. Then :

1. (1) $n=20$ and $\operatorname{Re}(z)=-10$

2. (2) $n=40$ and $\operatorname{Re}(z)=10$

3. (3) $n=40$ and $\operatorname{Re}(z)=-10$

4. (4) $n=20$ and $\operatorname{Re}(z)=10$

Correct Option: 3,

Solution:

Let $\operatorname{Re}(z)=x$ i.e., $z=x+10 i$

$2 z-n=(2 i-1)(2 z+n)$

$(2 x-n)+20 i=(2 i-1)((2 x+n)+20 i)$

On comparing real and imaginary parts,

$-(2 x+n)-40=2 x-n$ and $20=4 x+2 n-20$

$\Rightarrow 4 x=-40$ and $40=-40+2 n$

$\Rightarrow x=-10$ and $n=40$

Hence, $\operatorname{Re}(z)=-10$