Question

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

Answer 1

 Any line through the point of intersection of given lines is

The straight line meets x-axis at point A

The straight line meets the y-axis at point B

Let the midpoint of AB be P(h, k). Then

⇒ 2hk (a + b) = ab(h + k)

The locus of P(h, k) is 2xy(a + b) = ab(x + y).