Question

If $a,b$ are roots of quadratic equation $(2-b)x^2+\left(b^2-3b\right)x+b^2=0;b\ne 2;b\ne 0$ then

• A. a, b, ab are in A.P.
• B. $ab,a,b$ are in A.P.
• C. $2^{ab},2^a,2^b$ are in G.P.
• D. $\frac{1}{a},\frac{1}{b},\frac{1}{ab}$ are in H.P.

Answer 1

Answer: (B), (C)

$ab=\frac{b^2}{2-b}\Rightarrow a=\frac{b}{2-b}$
$\therefore a+1=\frac{2}{2-b}\Rightarrow (a+1)b=\frac{2b}{2-b}=2a$
$\Rightarrow ab+b=2a$
$\therefore ab,a,b\text{ are in A.P. }$
$\therefore 2^{ab},2^a,2^b\text{ are in G.P. }$