a, b, and c are all different and non-zero real numbers in arithmetic progression. If the roots of quadratic equation ax2+bx+c=0 are α and β such that 1α+1β,α+β and α2+β2 are in geometric progression, then the value of ac will be____.

We have (α+β)2=1α+1βα2+β2⇒ (α+β)2=1α+1β(α+β)2−2αβSubstituting  α+β=−ba and αβ=ca we have  b2a2=−bcb2a2−2cacb2+bb2−2ac=0b≠0, ∴bc+b2−2ac=0a,b,c are in AP,  ∴b=a+c2Therefore, we have (a+c)c2+a+c22−2ac=0⇒ a2−4ac+3c2=0  or (a−c)(a−3c)=0a≠c ∴a=3c ∴ac=3