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}$