177. Find the necessary and sufficient condition for the complex number $a+b i$ to be 1) equal to its conjugate, 2) equal to the reciprocal of its conjugate, 3) equal to the opposite of its conjugate number.
Instruction. 2. To

$$
a+b i=\frac{1}{a-b i}
$$

or

$$
(a+b i)(a-b i)=1
$$

the equality $a^{2}+b^{2}=1$ must hold. Conversely, if $a^{2}+b^{2}=1$, then $a+b i=\frac{1}{a-b i}$. Therefore, $a^{2}+b^{2}=1$ is the required condition.

Answer. 1) $b=0 ; 2) a^{2}+b^{2}=1$; 3) $a=0$.