JEE Main 2024 — 29 January, Shift 2 — Question 9

$\log _{\mathrm{e}} \mathrm{a}, \log _{\mathrm{e}} \mathrm{b}, \log _{\mathrm{e}} \mathrm{c}$ are in A.P.

$\therefore \mathrm{b}^{2}=\mathrm{ac}$ Also $\log _{e}\left(\frac{a}{2 b}\right), \log _{e}\left(\frac{2 b}{3 c}\right), \log _{e}\left(\frac{3 c}{a}\right)$ are in A.P.

$\left(\frac{2 b}{3 c}\right)^{2}=\frac{\mathrm{a}}{2 \mathrm{~b}} \times \frac{3 \mathrm{c}}{\mathrm{a}}$

$\frac{\mathrm{b}}{\mathrm{c}}=\frac{3}{2}$

Putting in eq. (i) $\mathrm{b}^{2}=\mathrm{a} \times \frac{2 \mathrm{~b}}{3}$

$$\begin{aligned}& \frac{a}{b}=\frac{3}{2} &\end{aligned}$$

$a: b: c=9: 6: 4$