$$\eqalign{ & {\text{Let }}\frac{{\text{a}}}{{\text{b}}}: - \frac{{\text{b}}}{{\text{a}}} = {\text{x}}:{\text{y}}{\text{.}} \cr & {\text{If}}\left( {{\text{x - y}}} \right) = \left\{ {\frac{{\text{a}}}{{\text{b}}}{\text{ + }}\frac{{\text{b}}}{{\text{a}}}} \right\}{\text{,}} \cr & {\text{then x is equal to - }} \cr}$$

Explanation:-

$$\eqalign{ & \Rightarrow \frac{x}{y} = \frac{{\left( {\frac{a}{b}} \right)}}{{\left( { - \frac{b}{a}} \right)}} = - \frac{{{a^2}}}{{{b^2}}} \cr & \Rightarrow y = \left( { - \frac{{{b^2}}}{{{a^2}}}} \right)x. \cr & \therefore x - y = \frac{{\text{a}}}{{\text{b}}}{\text{ + }}\frac{{\text{b}}}{{\text{a}}} \cr & \Rightarrow x + \frac{{{b^2}}}{{{a^2}}}x = \frac{{{a^2} + {b^2}}}{{ab}} \cr & \Rightarrow x\left( {\frac{{{a^2} + {b^2}}}{{{a^2}}}} \right) = \frac{{{a^2} + {b^2}}}{{ab}} \cr & \Rightarrow x = \frac{{{a^2}}}{{ab}} = \frac{a}{b}. \cr}$$