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