Question

( b=frac{2 a c}{a+c} )
( frac{L H S}{N O C O} frac{1}{b-a}+frac{1}{b-C}=frac{1}{left(frac{2 a C}{a+C}right)-a}+frac{1}{left(frac{2 a C}{a+C}right)} )
( =frac{a+c}{2 a c-a(a+c)}+frac{a+c}{2 a c-c(a+c)} )
( =frac{a+c}{a c-a^{2}}+frac{a+c}{a c-c^{2}} )
( Rightarrow frac{a+c}{a(c-a)}+frac{a+c}{c(a-c)} Rightarrow frac{a+c}{a(c-a)}+frac{a+c}{c(-c+a)} )
( Rightarrow quad frac{a+c}{a(c-a)}-frac{a+c}{c(c-a)} )
( Rightarrowleft(frac{a+c}{c-a}right)left[frac{1}{a}-frac{1}{c}right] Rightarrowleft(frac{a+c}{c-a}right) frac{x(c-a)}{a c} )
( Rightarrow quad frac{a+c}{a c} quad Rightarrow quad frac{a}{a c}+frac{c}{a c} )
( Rightarrow quad frac{1}{c} quad+frac{1}{a}=R H S )

# 13. If b is the harmonic mean between a and c, prove 1 1 1 1 that b -a b-C

Solution