An object has image thrice of its original size when kept at `8 cm` and `16 cm` from a convex lens. Focal length of the lens is

Options

A. `8 cm`
B. `16 cm`
C. Between `8 cm` and `16 cm`
D. Less than `8 cm`

Explanation

To find the focal length of the convex lens given the conditions of the problem, we can use the lens formula and the magnification formula.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have an object placed at two distances from a convex lens: \( u_1 = -8 \, \text{cm} \) (for virtual image) and \( u_2 = -16 \, \text{cm} \) (for real image).
- The magnification \( M \) is given as \( +3 \) for the virtual image and \( -3 \) for the real image.

2. **Using the Magnification Formula**:
- The magnification \( M \) is given by the formula:
\[ M = \frac{v}{u} \]
- Rearranging gives:
\[ v = M \cdot u \]

3. **Finding Image Distance for Virtual Image**:
- For the virtual image at \( u_1 = -8 \, \text{cm} \):
\[ M = +3 \implies v_1 = 3 \cdot (-8) = -24 \, \text{cm} \]
- Here, \( v_1 \) is negative, indicating a virtual image.

4. **Using the Lens Formula**:
- The lens formula is:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
- Substituting \( v_1 \) and \( u_1 \):
\[ \frac{1}{f} = \frac{1}{-24} - \frac{1}{-8} \]
- This simplifies to:
\[ \frac{1}{f} = -\frac{1}{24} + \frac{1}{8} \]
- Finding a common denominator (24):
\[ \frac{1}{f} = -\frac{1}{24} + \frac{3}{24} = \frac{2}{24} = \frac{1}{12} \]
- Therefore, \( f = 12 \, \text{cm} \).

5. **Finding Image Distance for Real Image**:
- For the real image at \( u_2 = -16 \, \text{cm} \):
\[ M = -3 \implies v_2 = -3 \cdot (-16) = 48 \, \text{cm} \]

6. **Using the Lens Formula Again**:
- Substituting \( v_2 \) and \( u_2 \):
\[ \frac{1}{f} = \frac{1}{48} - \frac{1}{-16} \]
- This simplifies to:
\[ \frac{1}{f} = \frac{1}{48} + \frac{1}{16} \]
- Finding a common denominator (48):
\[ \frac{1}{f} = \frac{1}{48} + \frac{3}{48} = \frac{4}{48} = \frac{1}{12} \]
- Thus, \( f = 12 \, \text{cm} \).

### Conclusion:
The focal length of the lens is \( f = 12 \, \text{cm} \).