Question

If $z_1=\frac{1}{a+i},a\ne 0$ and $z_2=\frac{1}{1+bi},b\ne 0$ are such that $z_1={\overline{z}}_2$ then

• A. $a=-1,b=1$
• B. $a=1,b=-1$
• C. $a=2,b=1$
• D. $a=1,b=2$

Answer 1

Answer: (B)

$a=1,b=-1$

Given $z_1=\frac{1}{a+i}=\frac{1}{a+i}\frac{a-i}{a-i}=\frac{a-i}{1+a^2}$
and $z_2=\frac{1}{1+bi}=\frac{1}{1+bi}\frac{1-bi}{1-bi}=\frac{1-bi}{1+b^2}$
Also given, $z_1=\overline{z_2}$
$\Rightarrow \frac{a-i}{1+a^2}=\frac{1+bi}{1+b^2}$
$\Rightarrow \frac{a}{1+a^2}=\frac{1}{1+b^2}$ and $\frac{-1}{1+a^2}=\frac{b}{1+b^2}$
By solving these equation we get $a=1,b=-1$