NDA2015MathematicsTrigonometric Ratios & IdentitiesActual
If x+ y=a and x+ y=b , then ^2 ( x+y 2 )+ ^2 ( x-y 2 ) is equal to
Options
- Aa^4+b^4+4 b^2 a^2 b^2+b^4
- Ba^4-b^4+4 b^2 a^2 b^2+b^4
- Ca^4-b^4+4 a^2 a^2 b^2+a^4
- DNone of the above
Correct answer
B. a^4-b^4+4 b^2 a^2 b^2+b^4
Step-by-step solution
aligned & x + y = a & 2 ( x + y 2 ) ( x - y 2 )= a ....(1) & x + y = b & 2 ( x + y 2 ) ( x - y 2 )= b ....(2) aligned dividing eq (1) & (2) ( x + y 2 )= a ~b Squaring eq (1) & (2) and adding - aligned & 4 ^2 ( x-y 2 )=a^2+b^2 & ^2 ( x-y 2 )= 4 a^2+b^2 aligned again- aligned & ^2 ( x + y 2 )+ ^2 ( x - y 2 ) = & ( a ~b )^2+ ^2 ( x - y 2 )-1 = & a ^2 ~b ^2 + 4 a ^2+ b ^2 -1 = & a ^4- b ^4+4 ~b ^2 a ^2 ~b ^2+ b ^4 aligned