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Question

Suppose that the three quadratic equations ax22bx+c=0, bx22cx+a=0 and cx22ax+b=0 all have only positive roots. Then

A
b2=ca
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B
c2=ab
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C
a2=bc
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D
a=b=c
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Solution

The correct option is D a=b=c
ax22bx+c=0, bx22cx+a=0 and cx22ax+b=0 all have positive roots only.
Product of roots >0
ca>0, ab>0, bc>0
a,b,c have same sign.

Also, Δ0
b2ac, c2ab, a2bc (1)
b2c2acab=a2bc, c2a2b2ac, a2b2c2ab
bca2, acb2, abc2 (2)

From (1) and (2),
a2=bc, b2=ac, c2=ab
a3=b3=c3=abc
a=b=c

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