As the roots of the given quadratic equation are equal and real then its discriminant will be zero
[b(c-a)]^2=4×a(b-c)×c(a-b)
=>(bc)^2+(ab)^2+2ab^2c=4[(ab-ca))(ca-bc)]
=>(bc)^2+(ab)^2-2ab^2c=4a^2bc-4ab^2c +4abc^2-4(ca)^2
=>(bc)^2+(ab)^2+(2ca)^2+2ab^2c-4abc^2-4a^2bc=0
=>(bc+ab-2ca)^2=0
=>bc+ab=2ca
=>1/a+1/c=2/b ( dividing both sides by abc)
Proved