Question

For two non-zero complex number $A$ & $B$ if $A+\frac{1}{B}=\overline{A}$ and $\frac{1}{A}+B=\overline{B}$ then-

• A. $A$ and $B$ are purely imaginary
• B. $A$ and $B$ are purely real
• C. $A=-\frac{1}{2}$
• D. $A=\frac{1}{2}$

Answer 1

The correct option is A $A$ and $B$ are purely imaginary

Solution -

Let $A=x_1+i{y}_1$

$B=x_2+i{y}_2$

$A+\frac{1}{B}=\overline{A}$

$x_1+i{y}_1+\frac{1}{x_2+i{y}_2}=x_1-i{y}_1$

$x_1+i{y}_1+\frac{x_2-i{y}_2}{x_2^2+y_2^2}=x_1-i{y}_1$

$\frac{x_2}{x_2^2+y_2^2}=0$.

$2y_1=\frac{y_2}{x_2^2+y_2^2}$

$\frac{1}{A}+B=\overline{B}$

similarly we will get,

$\frac{x_1}{x_1^2+y_1^2}=0$ $2y_2=\frac{y_1}{x_1^2+y_1^2}$

$\therefore x_1=0,x_2=0$

A & B are purely imaginary.

A is correct.