A point object is placed at a distance of `12 cm` from a convex lens of focal length `10 cm.` On the other side of the lens, a convex mirror is placed at a distance of `10 cm` from the lens such that the image formed by the combination coincides with the object itself. The focal length of the convex mirror is

Options

A. `20 cm`
B. `25 cm`
C. `15 cm`
D. `30 cm`

Explanation

To solve the problem step by step, we will use the lens formula and the properties of mirrors.

### Step 1: Identify the given values
- Distance of the object from the lens (u) = -12 cm (the negative sign indicates that the object is on the same side as the incoming light)
- Focal length of the convex lens (f) = +10 cm (positive for convex lens)
- Distance of the convex mirror from the lens = 10 cm

### Step 2: Use the lens formula to find the image distance (v)
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Rearranging the formula gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]

Substituting the values:
\[ \frac{1}{v} = \frac{1}{10} + \frac{1}{-12} \]
Calculating the right-hand side:
\[ \frac{1}{v} = \frac{1}{10} - \frac{1}{12} \]
To combine these fractions, find a common denominator (which is 60):
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
Thus,
\[ \frac{1}{v} = \frac{6}{60} - \frac{5}{60} = \frac{1}{60} \]
So,
\[ v = 60 \text{ cm} \]

### Step 3: Determine the distance of the image from the mirror
Since the convex mirror is placed 10 cm away from the lens, the distance of the image from the mirror (denoted as d) can be calculated as:
\[ d = v - 10 = 60 - 10 = 50 \text{ cm} \]

### Step 4: Use the mirror formula to find the focal length of the convex mirror
The mirror formula is given by:
\[ \frac{1}{f_m} = \frac{1}{v_m} + \frac{1}{u_m} \]
For the convex mirror:
- The image distance (v_m) is -50 cm (the negative sign indicates that the image is virtual and on the same side as the object).
- The object distance (u_m) is -10 cm (the object is virtual and on the same side as the incoming light).

Substituting these values into the mirror formula:
\[ \frac{1}{f_m} = \frac{1}{-50} + \frac{1}{-10} \]
Finding a common denominator (which is 50):
\[ \frac{1}{-50} + \frac{1}{-10} = \frac{-1}{50} + \frac{-5}{50} = \frac{-6}{50} \]
Thus,
\[ \frac{1}{f_m} = \frac{-6}{50} \implies f_m = -\frac{50}{6} = -8.33 \text{ cm} \]

### Step 5: Conclusion
The focal length of the convex mirror is approximately **8.33 cm** (the negative sign indicates that it is a convex mirror).