Suppose that the three quadratic equations ax2 –2bx+c = 0, bx2 –2cx+a = 0 and cx2 –2ax+b = 0 all have only positive roots. Then

Since the roots of the three equations are positive,all of a, b, c have the same sign.Also b2≥ac,c2≥ab and a2 ≥ bc .Hence b2c2≥a2(bc) which implies bc≥a2Therefore a2 = bc similarly b2 = ca and c2 = ab.Thus a = b = c.

Suppose that the three quadratic equations ax2 –2bx+c = 0, bx2 –2cx+a = 0 and cx2 –2ax+b = 0 all have only positive roots. Then

Since the roots of the three equations are positive,all of a, b, c have the same sign.Also b2≥ac,c2≥ab and a2 ≥ bc .Hence b2c2≥a2(bc) which implies bc≥a2Therefore a2 = bc similarly b2 = ca and c2 = ab.Thus a = b = c.

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Suppose that the three quadratic equations ax² –2bx+c = 0, bx² –2cx+a = 0 and cx² –2ax+b = 0 all have only positive roots. Then

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ab+bc+ca=0

a,b,c need not all be equal

a=b=c

abc = 1

answer is A.

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Detailed Solution

Since the roots of the three equations are positive,
all of a, b, c have the same sign.
Also $b^{2} \geq ac, c^{2} \geq ab$ and a² $\geq$ bc .
Hence $b^{2} c^{2} \geq a^{2} (bc)$ which implies $b c \geq a^{2}$
Therefore a² = bc similarly b² = ca and c² = ab.
Thus a = b = c.