Given: ax2+bx+c=0 is a quadratice equation Roots are in the ratio c:1 i.e α:β=c:1 or β:α=c:1 Concept Used: Sum of roots (α+β)=−b∕a Product of roots (α.β)=c∕a Calculation: Given α:β=c:1... (i) According to the question (α+β)=−b∕a⋯ (ii) (α.β)=c∕a⋯ (iii) On squaring the (i) equation, we get α2+β2+2α⋅β=
b2
a2
Dividing the above equation by α.β, we get ⇒
α2
α⋅β
+
β2
α⋅β
+2=
b2
a2
c
a
⇒
α
β
+
β
α
+2=
b2
ac
Now, from (i) ⇒c+
1
c
+2=
b2
ac
Multiplying the above equation by ac both side, we get ⇒ac2+a+2ac=b2 ⇒a(c2+1+2c)=b2 ⇒a(c+1)2=b2 ∴ The correct relation is b2=a(c+1)2.