Question

If $a,b,c$ are three distinct numbers in G.P., $b,c,a$ are in A.P. and $a,bc,abc$ are in H.P. then correct relation can be

• A. $a>b>c$
• B. $a>c>b$
• C. $b>c>a$
• D. $c>a>b$

Answer 1

Answer: (B), (C)

$a>c>b$
$b>c>a$

Let $a=\frac{A}{r},b=A$ and $C=Ar$
$\because$ $b,c,a$ are in A.P. $\Rightarrow 2c=a+b$
$\Rightarrow 2Ar=\frac{A}{r}+A\Rightarrow 2r^2-r-1=0$
$\Rightarrow r=1$ or $r=-\frac{1}{2}$
$\because$ $a,b,c$ are distinct $\Rightarrow r=-\frac{1}{2}$
$\therefore a=-2A,b=A,c=-\frac{A}{2}$
$a,bc,abc$ are in H.P. $\Rightarrow \frac{1}{a},\frac{1}{bc},\frac{1}{abc}$ are in A.P.
$\Rightarrow \frac{2}{bc}=\frac{1}{abc}+\frac{1}{a}\Rightarrow -\frac{4}{A^2}=\frac{1}{A^3}-\frac{1}{2A}$
$\Rightarrow -\frac{4}{A^2}=\frac{2-A^2}{2A^3}\Rightarrow -8A=2-A^2$
$\Rightarrow A^2-8A-2=0\Rightarrow$ $A$ has two values one is positive and other is negative.
if $A>0$ $\Rightarrow$ $b>c>a$.
if $A<0$ $\Rightarrow$ $a>c>b$.