Question

Solve using Componendo - Dividendo:

[ √(a+x) + √(a-x) ] / [ √(a+x) - √(a-x) ], when x = (2ab) / (b² +1)​

Answer 1

Answer:

Given: x=2abb2+1

a+x√+a−x√a+x√−a−x√

= a+x√+a−x√a+x√−a−x√×a+x√+a−x√a+x√+a−x√

= (a+x√)2+(a−x√)2+2a+x√.a−x√(a+x√)2−(a−x√)2

= a+x+a−x+2a2−x2√a+x−a+x

= 2a+2a2−x2√2x

= a+a2−x2√x

= a+a2−(2abb2+1)2√2abb2+1

= a+a2−4a2b2(b2+1)2−−−−−−−−−√×b2+12ab

= a+a(b2+1)2−4b2(b2+1)2−−−−−−−−√×b2+12ab

= a[1+1b2+1b4+1+2b2−4b2−−−−−−−−−−−−−−√]×b2+12ab

= b2+1+b4+1−2b2√b2+1×b2+12b

= b2+1+(b2−1)2√2b

= b2+1+b2−12b

= 2b22b

= b