If (x+y) `prop ` (x-y) , then prove that (ax +by) `prop` (px+qy) where a ,b ,p, q are non -zero variation constant.

Explanation

(x+y) `prop` (x-y) `rArr` x+y =k (x-y) , (where k `ne` 0= variation constant )
`(x+y)/(x-y) =k rArr (x+y+x-y)/(x+y-x+y)=(k+1)/(k-1)` (by componendo and dividendo )
`rArr (2x)/(2y)=(k+1)/(k-1)`
`rArr x/y=(k+1)/(k-1)`
` (ax)/(by)=(a(k+1))/(b(k-1))`
`rArr (ax+by)/(by) =(a(k+1)+b(k-1))/(b(k-1)).....(1)`
again, `x/y=(k+1)/(k-1)rArr (px)/(qy)=(p(k+1))/(q(k-1))`
` rArr (px+qy)/(qy)=(p(k+1)+q(k-1))/(q(k-1))......(2)`
Now , dividing (1)by(2) we get ,`((ax+by)/(by))/((px+qy)/(qy))=((a(k+1)+b(k-1))/(b(k-1)))/((p(k+1)+q(k-1))/(q(k-1))`
or, `(ax+by)/(px+qy)xxq/b=(a(k+1)+(b(k-1)))/(p(k+1)+q(k-1))xxq/b`
or, `(ax+by)/(px+qy)=(a(k+1)+(b(k-1)))/(p(k+1)+q(k-1))`
or, `(ax+by)/(px+qy)=(a(k+1)+(b(k-1)))/(p(k+1)+q(k-1))=m("let")`
`because` a, b, p, q, k are non- zero variation constant .
`therefore` m is also a non -zero constant .
`therefore (ax+by)/(px+qy)=m ne0`
`rArr ax +by=m (px+qy)`
`rArr (ax+by)prop (px+qy) (because m `=non -zero varaition cosntant ).
Hence (ax+by)`prop` (px+qy) . (proved)
(x+y) `prop (x-y) rArr x+y=k(x-y)`
`rArr (x+y)/(x-y)=k`
`rArr (x+y+x-y)/(x+y-x+y)=(k+1)/(k-1) ` [by componendo and dividendo]
`rArr (2x)/(2y)=m("let") [when m=(k+1)/(k-1)ne0] `
`rArr x/y =mrArr x=my`
Now , `(ax+by)/(px+qy)=(axxmy+by)/(pxxmy+qy)=(y(am+b))/(y("pm"+q))`
`=(am+b)/("pm"+q)=k("let")[when k =(am+b)/("pm"+q)ne0]`
`therefore (ax+by)/(px+qy)=k rArr ax +by=k (px+qy)`
`therefore (ax+by) prop (px+qy) (proved )[because k ne0= ` variation constant ]