First slide

Theory of equations

Question

Let three quadratic equations ${ax}^{2} - 2 bx + c = 0, {bx}^{2} - 2 cx + a = 0$ and ${cx}^{2} - 2 ax + b = 0$ , all have only positive roots. Then which of these are always true?

Moderate

A

b² = ac
B

c² = ab
C

each pair of equations has exactly one root common
D

each pair of equations has two roots common

Solution

For each equation, required conditions
(i) Discriminant $\geq 0$ $\Rightarrow b^{2} \geq ac, c^{2} \geq ab, a^{2} \geq bc$
(ii) $f (0) > 0 \Rightarrow c, a, b > 0$
(iii) Abscissa of vertex > 0 $\Rightarrow \frac{b}{a} > 0, \frac{c}{b} > 0, \frac{a}{c} > 0$ .
Now, $b^{2} \geq ac, c^{2} \geq ab \Rightarrow b^{2} c^{2} \geq a^{2} bc \Rightarrow bc \geq a^{2}$
But $a^{2} \geq bc . So, a^{2} = bc$ .
Similarly, b² = ac and c² = ab.
Therefore, a = b = c.

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