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