The equation of the line joining the origin to the point of intersection of the lines \(\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}} = 1\) and \(\frac{{\rm{x}}}{{\rm{b}}} + \frac{{\rm{y}}}{{\rm{a}}} = 1\) is

Calculation:Given LINES are \(\frac{{\rm{X}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{B}}} = 1\) and \(\frac{{\rm{x}}}{{\rm{b}}} + \frac{{\rm{y}}}{{\rm{a}}} = 1\)\(\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}} = 1\) ---(1)\(\frac{{\rm{x}}}{{\rm{b}}} + \frac{{\rm{y}}}{{\rm{a}}} = 1\) ---(2)Subtract EQUATION 1 - 2, we get\(\left( {\frac{{\rm{x}}}{{\rm{a}}} + \frac{{\rm{y}}}{{\rm{b}}}} \right) - \left( {\frac{{\rm{x}}}{{\rm{b}}} + \frac{{\rm{y}}}{{\rm{a}}}} \right) = 1 - 1\)\(\Rightarrow \left( {\frac{{\rm{x}}}{{\rm{a}}} - \frac{{\rm{x}}}{{\rm{b}}}} \right) + \left( {\frac{{\rm{y}}}{{\rm{b}}} - \frac{{\rm{y}}}{{\rm{a}}}} \right) = 0\)\(\Rightarrow {\rm{x}}\left( {\frac{1}{{\rm{a}}} - \frac{1}{{\rm{b}}}} \right) - {\rm{y}}\left( {\frac{1}{{\rm{a}}} - \frac{1}{{\rm{b}}}} \right) = 0\)\(\Rightarrow \left( {{\rm{x}} - {\rm{y}}} \right)\left( {\frac{1}{{\rm{a}}} - \frac{1}{{\rm{b}}}} \right) = 0\)∴ x - y = 0