Question

If $a,b,c$ are in $G.P.,x$ is the arithmetic mean between $a$ and $b$ and if $y$ is the arithmetic mean between $b$ and $c$ , then $\frac{1}{2x}+\frac{1}{2y}=$

• A. $\frac{1}{b}$
• B. $\frac{1}{a}$
• C. $\frac{1}{2}$
• D. $\frac{1}{a+b}$

Answer 1

Answer: (A)

According to question,
$b=\sqrt{ac}...(i)$
$a+b=2x...(ii)$
$b+c=2y...(iii)$
Now, $\frac{1}{2x}+\frac{1}{2y}=\frac{1}{a+b}+\frac{1}{b+c}$
$=\frac{a+c+2b}{\left(a+b\right)\left(b+c\right)}$
$=\frac{a+c+2\sqrt{ac}}{\left({\left(\sqrt{a}\right)}^2+\sqrt{ac}\right)\left(\sqrt{ac}+{\left(\sqrt{c}\right)}^2\right)}$
$=\frac{{\left(\sqrt{a}+\sqrt{c}\right)}^2}{{\left(\sqrt{a}+\sqrt{c}\right)}^2\left(\sqrt{ac}\right)}$
$=\frac{1}{\sqrt{ac}}=\frac{1}{b}$