Geo2.42

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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

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