Question

If a > 0, b > 0, c > 0 are positive read numbers in AP. If $a{x}^2+bx+c=0$ has real roots then

• A. $|\sqrt{\frac{a}{c}}-\sqrt{\frac{c}{a}}|\ge 2\sqrt{3}$
• B. $|\sqrt{\frac{a}{c}}-\sqrt{\frac{c}{a}}|\le 2\sqrt{3}$
• C. $|\sqrt{\frac{a}{c}}+\sqrt{\frac{c}{a}}|\ge 2\sqrt{3}$
• D. $|\sqrt{\frac{a}{c}}+\sqrt{\frac{c}{a}}|\le 2\sqrt{3}$

Answer 1

The correct option is A

$|\sqrt{\frac{a}{c}}-\sqrt{\frac{c}{a}}|\ge 2\sqrt{3}$

$a,b,c$ are in AP
$\Rightarrow 2b=a+c$
$b^2-4ac={\left(\frac{a+c}{2}\right)}^2-4ac=\frac{a^2+c^2+2ac}{4}-4ac=\frac{a^2+c^2-14ac}{4}$
real roots $\Rightarrow b^2-4ac\ge 0\Rightarrow a^2+c^2-14ac\ge 0.\Rightarrow a^2+c^2\ge 14ac\Rightarrow \frac{a}{c}+\frac{c}{a}-2\ge 14-2=={(\sqrt{\frac{a}{c}}-\sqrt{\frac{c}{a}})}^2\ge 12or|\sqrt{\frac{a}{c}}-\sqrt{\frac{c}{a}}|\ge 2\sqrt{3}.$