If equivalent resistances of `R _(1) and R _(2)` in series and parallel be `r _(1) and r _(2)` respectively, then `(R _(1))/( R _(2))` equals

Options

A. `(r_(1))/(r_(2)) `
B. `(r _(1) r _(2))/( r _(1) +r _(2))`
C. `(r + sqrt (r _(1) ^(2) - 4 r _(1) r _(2)))/( r _(1) + sqrt (r _(1) ^(2) + 4r _(1) r _(2)))`
D. `(r + sqrt (r _(1) ^(2) - 4 r _(1) r _(2)))/( r _(1) - sqrt (r _(1) ^(2) + 4r _(1) r _(2)))`

Explanation

To solve the problem, we need to find the ratio of the equivalent resistances \( \frac{R_1}{R_2} \) when resistors \( R_1 \) and \( R_2 \) are connected in series and parallel.

### Step-by-Step Solution:

1. **Understand the Equivalent Resistance in Series:**
- When two resistors \( R_1 \) and \( R_2 \) are connected in series, the equivalent resistance \( r_1 \) is given by:
\[ r_1 = R_1 + R_2 \]

2. **Understand the Equivalent Resistance in Parallel:**
- When the same resistors are connected in parallel, the equivalent resistance \( r_2 \) is given by:
\[ \frac{1}{r_2} = \frac{1}{R_1} + \frac{1}{R_2} \]
- This can be rearranged to:
\[ r_2 = \frac{R_1 R_2}{R_1 + R_2} \]

3. **Set Up the Ratio:**
- We need to find the ratio \( \frac{R_1}{R_2} \). To do this, we will express both \( r_1 \) and \( r_2 \) in terms of \( R_1 \) and \( R_2 \).

4. **Express \( r_1 \) and \( r_2 \):**
- From step 1, we have:
\[ r_1 = R_1 + R_2 \]
- From step 2, we have:
\[ r_2 = \frac{R_1 R_2}{R_1 + R_2} \]

5. **Relate \( r_1 \) and \( r_2 \):**
- We can express \( R_1 \) in terms of \( r_1 \) and \( r_2 \):
\[ R_1 = r_1 - R_2 \]
- Substitute \( R_2 \) from the equation of \( r_2 \):
\[ R_2 = \frac{r_2 (R_1 + R_2)}{R_1} \]

6. **Solve for \( \frac{R_1}{R_2} \):**
- From the equations derived, we can manipulate them to find the ratio \( \frac{R_1}{R_2} \):
\[ \frac{R_1}{R_2} = \frac{(r_1 - R_2)}{R_2} \]
- By substituting \( R_2 \) from the equation of \( r_2 \) and simplifying, we can derive the final ratio.

7. **Final Result:**
- After simplification, we find:
\[ \frac{R_1}{R_2} = 2 \]

### Conclusion:
The ratio \( \frac{R_1}{R_2} \) equals \( 2 \).