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 step by step, we can follow these instructions:

### Step 1: Understand the setup
We have a thin convex lens with a focal length of 30 cm placed in front of a plane mirror. An object is placed at a distance \( x \) from the lens, and we need to find the value of \( x \) such that the final image coincides with the object.

**Hint:** Visualize the arrangement of the lens and the mirror, and understand how light behaves when passing through a lens and reflecting off a mirror.

### Step 2: Identify the condition for the image to coincide with the object
For the final image to coincide with the object, the rays of light coming from the object must be parallel to the principal axis after passing through the lens. This occurs when the object is placed at the focal point of the lens.

**Hint:** Recall the properties of lenses, especially how they focus light. The focal point is crucial for determining where to place the object.

### Step 3: Determine the position of the object
Since the focal length of the convex lens is given as 30 cm, the object must be placed at the focal point of the lens for the rays to emerge parallel to the principal axis. Therefore, the distance \( x \) from the lens to the object should be equal to the focal length.

**Hint:** Remember that the focal length is the distance from the lens to the focal point, which is where the object needs to be placed.

### Step 4: Calculate the value of \( x \)
Given that the focal length \( f = 30 \) cm, we can conclude that:
\[ x = f = 30 \text{ cm} \]

**Hint:** Make sure to note the units when calculating distances in optics.

### Final Answer
Thus, the value of \( x \) is \( 30 \) cm.

### Summary of Steps:
1. Understand the lens-mirror setup.
2. Identify the condition for the image to coincide with the object.
3. Determine the position of the object based on the focal length.
4. Calculate the value of \( x \) as equal to the focal length.