Question

If $a^2+b^2,ab+bc$ and $b^2+c^2$ are in G.P., then $a,b,c$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is B G.P.

$a^2+b^2,ab+bc,b^2+c^2$ are in G.P.
$\Rightarrow (ab+bc{)}^2=(a^2+b^2\left)\right(b^2+c^2)$
$\Rightarrow a^2{b}^2+b^2{c}^2+2a{b}^{2}c=a^2{b}^2+a^2{c}^2+b^2{c}^2+b^4$
$\Rightarrow b^4+a^2{c}^2-2a{b}^{2}c=0$
$\Rightarrow (b^2-ac{)}^2=0$
$\Rightarrow b^2=ac$
$\Rightarrow a,b$ and $c$ are in G.P.