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A variable straight line drawn through the point of intersection of the straight lines {\frac  {x}{a}}+{\frac  {y}{b}}=1\, and {\frac  {x}{b}}+{\frac  {y}{a}}=1\, meets the coordinate axes at A and B. Show that the locus of the midpoint of AB is 2(a+b)xy=ab(x+y)\,

The point of intersection of the given two lines is \left({\frac  {ab}{a+b}},{\frac  {ab}{a+b}}\right)\,

Let P(x_{1},y_{1})\, be the midpoint of AB.

Therefore A=(2x_{1},0),B=(0,2y_{1})\,

Equation of AB is {\frac  {x}{2x_{1}}}+{\frac  {y}{2y_{1}}}=1\,

This line passes thro'the point of intersection, hence

{\frac  {ab}{2x_{1}(a+b)}}+{\frac  {ab}{2y_{1}(a+b)}}=1\,

Therefore,the locus of P is ab(x+y)=2xy(a+b)\,

Main Page:Geometry:Straight Lines-I