A variable line xa+yb=1 moves in such a way that harmonic mean of a  and  b is 8. Then the least area of triangle made by the line with the coordinate axes is

a, 8, b are in H.P ⇒1a+1b=1/4  ⇒b=4aa−4 ⇒ area A=12a×b=12×a×4a(a−4)=2a2a-4If f(a)=2a2a-4  f'(a)=a-44a-2a2(1)(a-4)2=0 2a2-16a=0f(a)  is minimum at  a=8  (given) . Hence minimum value of area is 2828-4=32 sq. units.