Question

Suppose that the three quadratic equations $a{x}^2-2bx+c=0,b{x}^2-2cx+a=0$ and $c{x}^2-2ax+b=0$ all have only positive roots. Then

• A. $b^2=ca$
• B. $c^2=ab$
• C. $a^2=bc$
• D. $a=b=c$

Answer 1

Answer: (A), (B), (C), (D)

$a=b=c$

$a{x}^2-2bx+c=0,b{x}^2-2cx+a=0$ and $c{x}^2-2ax+b=0$ all have positive roots only.
Product of roots $>0$
$\Rightarrow \frac{c}{a}>0,\frac{a}{b}>0,\frac{b}{c}>0$
$\Rightarrow a,b,c$ have same sign.

Also, $\Delta \ge 0$
$\Rightarrow b^2\ge ac,c^2\ge ab,a^2\ge bc\cdots \left(1\right)$
$\Rightarrow b^2{c}^2\ge ac\cdot ab=a^{2}bc,c^2{a}^2\ge b^{2}ac,a^2{b}^2\ge c^{2}ab$
$\Rightarrow bc\ge a^2,ac\ge b^2,ab\ge c^2\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right),$
$a^2=bc,b^2=ac,c^2=ab$
$\Rightarrow a^3=b^3=c^3=abc$
$\Rightarrow a=b=c$