An object is placed 42 cm, in front of a concave mirror of focal length 21 cm. Light from the
concave mirror is reflected onto a small plane mirror 21 cm in front of the concave mirror
Where is the final image?

Explanation

To find the final image position when an object is placed in front of a concave mirror and light is reflected onto a plane mirror, we can follow these steps:

### Step 1: Identify the given values
- Object distance (u) from the concave mirror = -42 cm (the negative sign indicates that the object is in front of the mirror).
- Focal length (f) of the concave mirror = -21 cm (the negative sign indicates that it is a concave mirror).

### Step 2: Use the mirror formula to find the image distance (v)
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{-21} = \frac{1}{v} + \frac{1}{-42} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{-21} + \frac{1}{42} \]
Finding a common denominator (which is 42):
\[ \frac{1}{v} = \frac{-2 + 1}{42} = \frac{-1}{42} \]
Thus:
\[ v = -42 \text{ cm} \]
This means the image is formed at a distance of 42 cm in front of the concave mirror.

### Step 3: Position of the plane mirror
The plane mirror is placed 21 cm in front of the concave mirror. Since the image formed by the concave mirror is at 42 cm in front of it, the distance from this image to the plane mirror is:
\[ 42 \text{ cm} - 21 \text{ cm} = 21 \text{ cm} \]

### Step 4: Image formation by the plane mirror
For a plane mirror, the image is formed at the same distance behind the mirror as the object is in front of it. Therefore, the image formed by the plane mirror will be:
\[ 21 \text{ cm} \text{ behind the plane mirror} \]
Since the plane mirror is 21 cm in front of the concave mirror, the final image position relative to the concave mirror is:
\[ 21 \text{ cm (behind the plane mirror)} + 21 \text{ cm (in front of the concave mirror)} = 42 \text{ cm} \]

### Conclusion
The final image is located at 42 cm in front of the concave mirror, which is at the center of curvature of the concave mirror.

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