Two mirrors, one concave and the other convex, are placed 60 cm apart with their reflecting surfaces facing each other. An object is placed 30 cm from the pole of either of them on their axis. If the focal lengths of both the mirrors are 15 cm, the position of the image formed by reflection, first at the convex and then at the concave mirror, is:

Options

A. 19.09 cm from the pole of the concave mirror
B. 19.09 cm from the pole of the convex mirror
C. 11.09 cm from the pole of the concave mirror
D. 11.09 cm from the pole of the convex mirror

Explanation

To solve the problem of finding the position of the image formed by the concave and convex mirrors, we will follow these steps:

### Step 1: Identify the Object Distance for the Convex Mirror
The object is placed 30 cm from the pole of the convex mirror. According to the sign convention, we take the object distance (u) as negative for mirrors. Thus, we have:
\[ u_1 = -30 \, \text{cm} \]

### Step 2: Identify the Focal Length of the Convex Mirror
The focal length (f) of the convex mirror is given as +15 cm (positive because the focus of a convex mirror is virtual and located behind the mirror).

### Step 3: Use the Mirror Formula for the Convex Mirror
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging gives:
\[ \frac{1}{v_1} = \frac{1}{f} - \frac{1}{u_1} \]

Substituting the values:
\[ \frac{1}{v_1} = \frac{1}{15} - \frac{1}{-30} \]
\[ \frac{1}{v_1} = \frac{1}{15} + \frac{1}{30} \]

Finding a common denominator (30):
\[ \frac{1}{v_1} = \frac{2}{30} + \frac{1}{30} = \frac{3}{30} = \frac{1}{10} \]

Thus,
\[ v_1 = 10 \, \text{cm} \]

### Step 4: Determine the Position of the Image from the Convex Mirror
The positive value of \( v_1 \) indicates that the image is formed 10 cm behind the convex mirror.

### Step 5: Identify the Object Distance for the Concave Mirror
The distance between the two mirrors is 60 cm. The image formed by the convex mirror (I1) is 10 cm behind it, which means it is located at:
\[ 60 \, \text{cm} - 10 \, \text{cm} = 50 \, \text{cm} \]
from the pole of the concave mirror. Therefore, the object distance for the concave mirror (u2) is:
\[ u_2 = -50 \, \text{cm} \]

### Step 6: Identify the Focal Length of the Concave Mirror
The focal length (f) of the concave mirror is given as -15 cm (negative because the focus of a concave mirror is real and located in front of the mirror).

### Step 7: Use the Mirror Formula for the Concave Mirror
Using the mirror formula again:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging gives:
\[ \frac{1}{v_2} = \frac{1}{f} - \frac{1}{u_2} \]

Substituting the values:
\[ \frac{1}{v_2} = \frac{1}{-15} - \frac{1}{-50} \]
\[ \frac{1}{v_2} = -\frac{1}{15} + \frac{1}{50} \]

Finding a common denominator (150):
\[ \frac{1}{v_2} = -\frac{10}{150} + \frac{3}{150} = -\frac{7}{150} \]

Thus,
\[ v_2 = -\frac{150}{7} \approx -21.43 \, \text{cm} \]

### Step 8: Determine the Position of the Final Image
The negative value of \( v_2 \) indicates that the final image is formed 21.43 cm in front of the concave mirror.

### Final Answer
The position of the final image formed by the combination of the two mirrors is approximately 21.43 cm in front of the concave mirror.

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