Question

If three unequal positive real numbers a,b,c are in G.P. and a-b,c-a, a-b are in H.P., then the values of a+b+c is independent of

• A. a
• B. b
• C. c
• D. None of these

Answer 1

The correct option is D

None of these

As $a$, $b$, $c$ are in G.P, $b^2$ = ac and $b-c$, $c-a$, $a-b$ are in H.P.
$\frac{2}{c-a}=\frac{1}{b-a}+\frac{1}{a-b}=\frac{a-c}{(b-c)(a-b)}$
$\Rightarrow 2(b-c)(a-b)=-(a-c{)}^2\Rightarrow 2(ab-ac-b^2-bc)=-(\sqrt{a}-\sqrt{c}{)}^2(\sqrt{a}+\sqrt{c}{)}^{2}2(ab-2b^2+bc)=-(\sqrt{a}-\sqrt{c}{)}^2(\sqrt{a}+\sqrt{c}{)}^2$
$\Rightarrow 2b(\sqrt{a}-\sqrt{c}{)}^2=-(\sqrt{a}-\sqrt{c}{)}^2(\sqrt{a}+\sqrt{c}{)}^2$ $[\because \sqrt{a}-\sqrt{c}\ne 0]$
$2b=(a+c+2\sqrt{ac})=-(a+c+2b)\Rightarrow a+b+c=-3b=-3\sqrt{ac}$ which is not independent a, b and c.
$\therefore$ Choice (4) is correct.