If H be the harmonic mean between x and y, then show that `(H+x)/(H-x)+(H+y)/(H-y)=2`

Explanation

We have, `H=(2xy)/(x+y)`
`therefore (H)/(x)=(2xy)/(x+y) " and "(H)/(y)=(2x)/(x+y)`
By componendo and dividendo, we have
`(H+x)/(H-x)=(2y+x+y)/(2y-x-y)=(x+3y)/(y-x)`
and `(H+y)/(H-y)=(2x+x+y)/(2x-x-y)=(3x+y)/(x-y)`
` therefore (H+x)/(H-x)+(H+y)/(H-y)=(x+3y)/(y-x)+(3x+y)/(x-y)`
` = (x+3y-3x-y)/(y-x)=(2(y-x))/(y-x)=2`
Aliter `(H+x)/(H-x)+(H+y)/(H-y)=2`
` implies ((H+x)/(H-x)-1)=(1-(H+y)/(H-y)) implies (2x)/(H-x)=(-2y)/(H-y)`
`i.e.Hx-xy=-Hy+xy implies H(x+y)=2xy`
`i.e.H =(2xy)/(x+y)`
which is true as, `x,H,y` are in HP. Hence, the required result.