A thin convex lens of focal length `30 cm` is placed in front of a plane mirror. An object is placed at a distance x from the lens (not in between lens and mirror ) so that its final image coincides with itself. Then, the value of x is

Options

A. `15 cm`
B. `30 cm`
C. `60 cm`
D. Insufficient data

Explanation

To solve the problem, we need to find the distance \( x \) from the lens to the object such that the final image coincides with the object itself.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a thin convex lens with a focal length \( f = 30 \, \text{cm} \).
- The lens is placed in front of a plane mirror.
- An object is placed at a distance \( x \) from the lens.

2. **Condition for Coinciding Image and Object**:
- For the final image to coincide with the object, the light rays must retrace their path after reflecting from the mirror.
- This can happen if the light rays from the object strike the mirror perpendicularly.

3. **Using the Lens Formula**:
- The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- \( f \) is the focal length,
- \( v \) is the image distance from the lens,
- \( u \) is the object distance from the lens (which is \( -x \) since the object is on the same side as the incoming light).

4. **Setting Up the Equation**:
- Since we want the image to coincide with the object, the image formed by the lens must be at the same distance as the object from the lens after reflection.
- If the object is at distance \( x \), the image formed by the lens (before reflection) will be at distance \( v \) from the lens.
- For the light to retrace its path, the distance from the lens to the mirror must equal the focal length. Therefore, the object must be placed at the focal length of the lens.

5. **Conclusion**:
- The object must be placed at a distance equal to the focal length of the lens from the lens itself.
- Thus, \( x = f = 30 \, \text{cm} \).

### Final Answer:
The value of \( x \) is \( 30 \, \text{cm} \).

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