Question

If $a,b\&3c$ are in arithmetic progression and $a,b\&4c$ are in geometric progression, then the possible values of $\frac{a}{b}$ are

• A. $\left\{\frac{2}{3},2\right\}$
• B. $\left\{\frac{3}{2},\frac{1}{2}\right\}$
• C. $\left\{\frac{2}{3},\frac{3}{2}\right\}$
• D. $\left\{\frac{1}{2},2\right\}$

Answer 1

Answer: (B)

$a+3c=2b$ and $b^2=4ac$
$\Rightarrow c=\frac{2b-a}{3}=\frac{b^2}{4a}$
$\therefore$ $8ab-4a^2=3b^2\Rightarrow 4a^2-8ab+3b^2=0$
$4\frac{a^2}{b^2}-8\frac{a}{b}+$ 3= $0\Rightarrow \frac{a}{b}=\frac{8\pm \sqrt{64-4\times 4\times 3}}{2\times 4}$
$\frac{a}{b}=\frac{2\pm \sqrt{4-3}}{2}=\frac{2\pm 1}{2}=\frac{3}{2},\frac{1}{2}.$