Three resistors having resistances r₁ , r₂ and r₃ are connected as shown in the given circuit. The ratio i₃/i₁ of currents in terms of resistances used in the circuit is :

Correct Option: The correct answer is r₂ / (r₂ + r₃) (which corresponds to the first option: r₂ / r₂ + r₃).

Step-by-Step Explanation:

1. In a typical electrical circuit containing three resistors, resistor r₁ is connected in series with a parallel combination of resistors r₂ and r₃.
The total current flowing through the main branch (and through resistor r₁) is designated as i₁.

2. When current i₁ reaches the parallel junction of r₂ and r₃, it divides into two branch currents:
- Current i₂ flows through resistor r₂.
- Current i₃ flows through resistor r₃.

3. Since the voltage across parallel branches is equal, we can write:
$v_{\text{parallel}}=i_2{r}_2=i_3{r}_3$

4. According to Kirchhoff's Current Law (KCL) at the junction:
$i_1=i_2+i_3$

5. Expressing i₂ in terms of i₃ using the parallel voltage relation:
$i_2=i_3\left(\frac{r_3}{r_2}\right)$

6. Substituting this expression into the KCL equation:
$i_1=i_3\left(\frac{r_3}{r_2}\right)+i_3$
Factor out i₃:
$i_1=i_3\left(\frac{r_3}{r_2},+,1\right)=i_3\left(\frac{r_3+r_2}{r_2}\right)$

7. To find the ratio of the currents i₃/i₁, rearrange the equation:
$\frac{i_3}{i_1}=\frac{r_2}{r_2+r_3}$