A point object is placed at a distance of 15 cm from a convex lens. The image is formed on the other side at a distance of 30 cm from the lens. When a concave lens is placed in contact with the convex lens, the image shifts away further by 30 cm. Calculate the focal lengths of the two lenses.

Explanation

To solve the problem step by step, we will first analyze the situation with the convex lens and then include the concave lens to find the focal lengths of both lenses.

### Step 1: Analyze the convex lens situation
1. **Given Data for Convex Lens:**
- Object distance (u) = -15 cm (the object distance is negative as per the sign convention)
- Image distance (v) = +30 cm (the image distance is positive as the image is formed on the opposite side of the lens)

2. **Using the Lens Formula:**
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{f_1} = \frac{1}{30} - \frac{1}{-15} \]
\[ \frac{1}{f_1} = \frac{1}{30} + \frac{1}{15} \]

3. **Finding a common denominator:**
The least common multiple of 30 and 15 is 30. Thus:
\[ \frac{1}{f_1} = \frac{1}{30} + \frac{2}{30} = \frac{3}{30} \]
\[ \frac{1}{f_1} = \frac{1}{10} \]

4. **Calculating the focal length of the convex lens:**
\[ f_1 = 10 \text{ cm} \]

### Step 2: Analyze the situation with the concave lens
1. **New Object and Image Distances:**
- The image formed by the convex lens acts as the object for the concave lens.
- Object distance for the concave lens (u₂) = -30 cm (the image from the convex lens is now the object for the concave lens)
- The final image is formed at a distance of 60 cm from the lens (v₂ = +60 cm).

2. **Using the Lens Formula for the Concave Lens:**
\[ \frac{1}{f_2} = \frac{1}{v_2} - \frac{1}{u_2} \]
Substituting the values:
\[ \frac{1}{f_2} = \frac{1}{60} - \frac{1}{-30} \]
\[ \frac{1}{f_2} = \frac{1}{60} + \frac{1}{30} \]

3. **Finding a common denominator:**
The least common multiple of 60 and 30 is 60. Thus:
\[ \frac{1}{f_2} = \frac{1}{60} + \frac{2}{60} = \frac{3}{60} \]
\[ \frac{1}{f_2} = \frac{1}{20} \]

4. **Calculating the focal length of the concave lens:**
\[ f_2 = -20 \text{ cm} \]

### Final Results
- Focal length of the convex lens \( f_1 = 10 \text{ cm} \)
- Focal length of the concave lens \( f_2 = -20 \text{ cm} \)