Two fixed points `A` and `B` are taken on the coordinates axes such that `O A=a` and `O B=b` . Two variable points `A '` and `B '` are taken on the same axes such that `O A^(prime)+O B^(prime)=O A+O Bdot` Find the locus of the point of intersection of `A B^(prime)` and `A^(prime)Bdot`

Answer

`(x)/(aa')+(y)/(b b')= 0`

Explanation

Let `A-= (a,0),B-= (0,b), A'-=(a',0), "and "B'-=(0,b').`
The equation of A'B is
`(x)/(a') + (y)/(b) = 1 " " (1)`
and the equation of AB' is
`(x)/(a) + (y)/(b') = 1 " "(2) `
Subtracting (1) from (2), we get
`x((1)/(a)- (1)/(a')) +y((1)/(b')- (1)/(b)) = 0`
`"or "(x(a'- a))/(aa') + (y(b- b'))/(b b') = 0`
`"or "(x)/(aa') + (y)/(b b') = 0 " " ["Using"a'-a=b-b']`