To find the focal length of a convex mirror , a student records the following data:
`{:("Object Pin","Convex Lens","Convex mirror","Image Pin"),(22.2 cm,32.2 cm,45.8 cm,71.2 cm):}`
The focal length of the convex lens is `f_(1)` and that of mirror is `f_(2)` . Then taking index correction to be negligibly small , `f_(1)` and `f_(2)` are close to :

Options

A. `f_(1) = 7.8 cm " " f_(2) = 12.7 cm`
B. `f_(1) = 15.6 cm " " f_(2) = 25.6 cm`
C. `f_(1) = 12.7 cm " " f_(2) = 7.8 cm`
D. `f_(1) = 7.8cm " " f_(2) = 25.4 cm`

Answer

To find the focal lengths of the convex lens and the convex mirror based on the given data, we can follow these steps:

### Step 1: Identify the given data
- Object distance for the convex lens (u) = 32.2 cm - 22.2 cm = 10 cm (but it will be negative)
- Image distance for the convex lens (v) = 71.2 cm - 32.2 cm = 39 cm
- Object distance for the convex mirror (u') = 22.2 cm - 45.8 cm = -23.6 cm (since it's virtual)
- Image distance for the convex mirror (v') = 71.2 cm - 45.8 cm = 25.4 cm

### Step 2: Calculate the focal length of the convex lens (f1)
Using the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{f_1} = \frac{1}{39} - \frac{1}{-10} \]
Calculating:
\[ \frac{1}{f_1} = \frac{1}{39} + \frac{1}{10} \]
Finding a common denominator (390):
\[ \frac{1}{f_1} = \frac{10}{390} + \frac{39}{390} = \frac{49}{390} \]
Thus,
\[ f_1 = \frac{390}{49} \approx 7.8 \text{ cm} \]

### Step 3: Calculate the focal length of the convex mirror (f2)
The radius of curvature (R) is given by:
\[ R = 25.4 \text{ cm} \]
The focal length of a convex mirror is given by:
\[ f_2 = \frac{R}{2} = \frac{25.4}{2} = 12.7 \text{ cm} \]

### Step 4: Final results
Thus, the focal lengths are:
- Focal length of convex lens (f1) = 7.8 cm
- Focal length of convex mirror (f2) = 12.7 cm

### Conclusion
The values of \( f_1 \) and \( f_2 \) are approximately 7.8 cm and 12.7 cm, respectively.

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