It is given that u^2 = v^2 + (m/n)((u-v)^2) and we have to prove that : v/u = (m-n)/(m+n)

u^2 = v^2 + (m/n)((u-v)^2)

=> u^2 - v^2 = (m/n)((u-v)^2)

=> (u^2 - v^2)/ (u-v)^2 = (m/n)

=> (u - v) ( u + v) / ( u - v)^2 = (m/n)

=> (u + v) / (u - v) = m/n

=> (u + v) / (u - v) + 1 = (m/n) + 1

=> (u + v + u - v) / (u - v) = (m + n) / n

=> 2u / ( u- v) = (m + n) / n

=> (u - v) / 2u = n/ ( m+ n)

=> (u - v) / u = 2n/ (m + n)

=> u / u - v / u = 2n / ( m + n)

=> 1 - v / u = 2n / ( m + n)

=> v / u = 1 - 2n / ( m + n)

=> v / u = (m + n - 2n ) / ( m+n)

=> v/ u = (m - n) / ( m + n)

**We prove that v/u = (m - n)/( m + n) if u^2 = v^2 + (m/n)((u-v)^2)**

We'll subtract v^2 both sides:

u^2 - v^2 = (u-v)(u+v)

We'll write (u-v)^2 = (u-v)(u-v)

We'll re-write the given expression:

(u-v)(u+v) = m(u-v)(u-v)/n

We'll simplify both sides:

u+v = m(u-v)/n

We'll multiply by n:

nu + nv = mu - mv

We'll move the terms in u to the left side and the terms in v to the right side:

nu - mu = -mv - nv

u(n-m) = -v(m+n)

u(m-n) = v(m + n)

We'll divide by u:

v(m + n)/u = m-n

We'll divide by m+n:

**v/u = (m-n)/(m+n) q.e.d.**