If the radius of the 3rd Bohr's orbit of hydrogen atom is r3 and the radius of 4th Bohr's orbit is r4, then:

🧠 The Geometric scaling In the Bohr model for Hydrogen, the orbit radius ($r_n$) depends purely on the square of the shell number ($n$). We are looking for the algebraic relationship between the 4th and 3rd orbits.

🗺️ The Proportion Mapping

1. General Rule: $r_n = n^2 \cdot a_0$.
2. For n=3: $r_3 = 3^2 \cdot a_0 = 9a_0 \implies a_0 = r_3/9$.
3. For n=4: $r_4 = 4^2 \cdot a_0 = 16a_0$.
4. Substitution: $r_4 = 16 \left( \frac{r_3}{9} \right) = \frac{16}{9} r_3$

⚡ The Direct Power Rule Since $r \propto n^2$, the ratio is simply the ratio of squares: $\frac{r_4}{r_3} = \frac{4^2}{3^2} = \frac{16}{9}$ Write this down once, and you have $r_4 = (16/9)r_3$ without even thinking about $a_0$.

⚠️ Common Traps The common "n ratio" mistake—don't write $4/3$. The relationship is square ($n^2$), not linear ($n$). Always pause to ensure you squared the numbers before picking the option.

$\boxed{\text{Answer: (b)}}$