Correct option is: (3) 18
\(\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)+\operatorname{Re}\left(\frac{\bar{z}-1}{2 \bar{z}-i}\right)=2\)
Here, \(\frac{\mathrm{z}-1}{2 \mathrm{z}+\mathrm{i}}=\left(\frac{\overline{\bar{z}-1}}{2 \bar{z}-\mathrm{i}}\right)=2\)
\(=\operatorname{Re}\left(\frac{\mathrm{z}-1}{2 \mathrm{z}+\mathrm{i}}\right)+\operatorname{Re}\left(\overline{\frac{\mathrm{z}-1}{2 \mathrm{z}+\mathrm{i}}}\right)=2\)
\(=2 \operatorname{Re}\left(\frac{z-1}{2 z+1}\right)=2 \Rightarrow \operatorname{Re}\left(\frac{z-1}{2 z+i}\right)=1\)
Let z = x + iy
\(\operatorname{Re}\left(\frac{(x-1)+i y}{2 x+i(2 y+1)}\right)=1 \Rightarrow \operatorname{Re}\left[\frac{((x-1)+i y)(2 x-i(y+1)}{(2 x+i(2 y+1)(2 x-i(2 y+1))}\right]=1\)
\(\Rightarrow \frac{2 \mathrm{x}(\mathrm{x}-1)+\mathrm{y}(2 \mathrm{y}+1)}{4 \mathrm{x}^{2}+(2 \mathrm{y}+1)^{2}}=1\)
\(\Rightarrow 2 \mathrm{x}^{2}-2 \mathrm{x}+2 \mathrm{y}^{2}+\mathrm{y}=4 \mathrm{x}^{2}+4 \mathrm{y}^{2}+1+4 \mathrm{y}\)
\(\Rightarrow 2 \mathrm{x}^{2}+2 \mathrm{y}^{2}+3 \mathrm{y}+2 \mathrm{x}+1=0\)
\(\Rightarrow x^{2}+y^{2}+x+\frac{3}{2} y+\frac{1}{2}=0\)
centre \(=\left(\frac{-1}{2}, \frac{-3}{4}\right), \mathrm{r}=\sqrt{\frac{1}{4}+\frac{9}{16}-\frac{1}{2}}=\frac{\sqrt{5}}{4}\)
\(\mathrm{a}=\frac{-1}{2}, \mathrm{~b}=\frac{-3}{4}, \mathrm{r}^{2}=\frac{5}{16}\)
\(15 \frac{\mathrm{ab}}{\mathrm{r}^{2}}=15 \times\left(\frac{-1}{2}\right) \times\left(\frac{-3}{4}\right) \times \frac{16}{5}=18\)
