Arrange the following combinations in the increasing order of focal length
a) Two piano convex lenses of focal lengths 20 cm and 30 cm in contact
b) Two convex lens of focal lengths 20 cm and 10 cm in contact
Two convex lenses of focal length 25 cm separated by 5 cm.

Options

A. a, b, c
B. b, a, c
C. a, c, b
D. c, b, a

Explanation

To solve the problem of arranging the combinations of lenses in increasing order of focal length, we will calculate the effective focal length for each combination step by step.

### Step 1: Calculate the focal length for combination (a)
**Combination (a)**: Two plano-convex lenses with focal lengths of 20 cm and 30 cm in contact.

Using the formula for the focal length of two lenses in contact:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
where \(f_1 = 20 \, \text{cm}\) and \(f_2 = 30 \, \text{cm}\).

Substituting the values:
\[ \frac{1}{F} = \frac{1}{20} + \frac{1}{30} \]
Finding the LCM of 20 and 30, which is 60:
\[ \frac{1}{F} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} \]
Thus,
\[ F = \frac{60}{5} = 12 \, \text{cm} \]

### Step 2: Calculate the focal length for combination (b)
**Combination (b)**: Two convex lenses with focal lengths of 20 cm and 10 cm in contact.

Using the same formula:
\[ \frac{1}{F_b} = \frac{1}{20} + \frac{1}{10} \]
Finding the LCM of 20 and 10, which is 20:
\[ \frac{1}{F_b} = \frac{1}{20} + \frac{2}{20} = \frac{3}{20} \]
Thus,
\[ F_b = \frac{20}{3} \approx 6.67 \, \text{cm} \]

### Step 3: Calculate the focal length for combination (c)
**Combination (c)**: Two convex lenses with focal lengths of 25 cm each, separated by 5 cm.

Using the formula for two lenses separated by a distance \(d\):
\[ \frac{1}{F_c} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} \]
Here, \(f_1 = 25 \, \text{cm}\), \(f_2 = 25 \, \text{cm}\), and \(d = 5 \, \text{cm}\).

Substituting the values:
\[ \frac{1}{F_c} = \frac{1}{25} + \frac{1}{25} - \frac{5}{25 \times 25} \]
Calculating:
\[ \frac{1}{F_c} = \frac{2}{25} - \frac{5}{625} \]
Finding the LCM of 25 and 625, which is 625:
\[ \frac{1}{F_c} = \frac{50}{625} - \frac{5}{625} = \frac{45}{625} \]
Thus,
\[ F_c = \frac{625}{45} \approx 13.89 \, \text{cm} \]

### Step 4: Arrange the focal lengths in increasing order
Now we have:
- \(F_a = 12 \, \text{cm}\)
- \(F_b \approx 6.67 \, \text{cm}\)
- \(F_c \approx 13.89 \, \text{cm}\)

Arranging these values in increasing order:
1. \(F_b \approx 6.67 \, \text{cm}\)
2. \(F_a = 12 \, \text{cm}\)
3. \(F_c \approx 13.89 \, \text{cm}\)

### Final Answer
The increasing order of focal lengths is:
**(b), (a), (c)**