A convex and a concave mirror of radii 10 cm are placed facing each other and 15 cm apart. An object is placed exactly between them. If the reflection first takes place in concave and then in convex mirror the position of the final image will be

Options

A. 7 cm behind concave mirror
B. at the pole of the concave mirror
C. at the pole of the convex mirror
D. 6.7 cm in front of concave mirror

Explanation

To solve the problem step by step, we will follow the principles of ray optics, specifically the mirror formula and the behavior of concave and convex mirrors.

### Step 1: Understand the Setup
We have a concave mirror and a convex mirror, both with a radius of curvature (R) of 10 cm. The distance between the two mirrors is 15 cm. The focal length (F) of a mirror is given by the formula:
\[ F = \frac{R}{2} \]
Thus, for both mirrors:
- Focal length of the concave mirror, \( F_1 = -5 \, \text{cm} \) (negative because it is concave)
- Focal length of the convex mirror, \( F_2 = +5 \, \text{cm} \) (positive because it is convex)

### Step 2: Determine the Object Distance for the Concave Mirror
The object is placed exactly between the two mirrors. The distance from the concave mirror to the object (U) can be calculated as:
\[ U = \frac{15 \, \text{cm}}{2} = 7.5 \, \text{cm} \]
Since the object is in front of the concave mirror, we take:
\[ U = -7.5 \, \text{cm} \]

### Step 3: Use the Mirror Formula for the Concave Mirror
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values for the concave mirror:
\[ \frac{1}{-5} = \frac{1}{v} + \frac{1}{-7.5} \]

### Step 4: Solve for the Image Distance (V) from the Concave Mirror
Rearranging the equation:
\[ \frac{1}{v} = \frac{1}{-5} + \frac{1}{7.5} \]
Finding a common denominator (which is 15):
\[ \frac{1}{v} = \frac{-3 + 2}{15} = \frac{-1}{15} \]
Thus:
\[ v = -15 \, \text{cm} \]
This means the image formed by the concave mirror is 15 cm in front of the mirror (on the same side as the object).

### Step 5: Determine the Object Distance for the Convex Mirror
The image formed by the concave mirror acts as the object for the convex mirror. The distance from the convex mirror to this image is:
\[ \text{Distance from convex mirror} = 15 \, \text{cm} - 15 \, \text{cm} = 0 \, \text{cm} \]
Since the image is at the pole of the convex mirror, the object distance for the convex mirror (U') is:
\[ U' = 0 \, \text{cm} \]

### Step 6: Use the Mirror Formula for the Convex Mirror
Now we apply the mirror formula for the convex mirror:
\[ \frac{1}{5} = \frac{1}{v'} + \frac{1}{0} \]
Since the object distance is 0, the image will be formed at the pole of the convex mirror.

### Final Answer
The final image after reflection from the convex mirror will be at the pole of the convex mirror, which is 0 cm from the mirror.