A concave mirror of focal length 10cm and a convex mirror of focal length 15cm are placed facing each other 40cm apart. A point object is placed between the mirrors, on their common axis and 15cm from the concave mirror. Find the position and nature of the image produced by successive reflections, first at the concave mirror and then at the convex mirror.

Options

A. 12cm behind convex mirror, real
B. 9cm behind convex mirror, real
C. 6cm behind convex mirror, virtual
D. 3cm behind convex mirror, virtual

Explanation

To solve the problem step by step, we will analyze the situation involving the concave and convex mirrors, and calculate the image positions and their nature.

### Step 1: Identify the Object Distance for the Concave Mirror
The object is placed 15 cm from the concave mirror. In the mirror formula, we take the object distance \( u \) as negative for concave mirrors. Therefore:
\[ u_1 = -15 \, \text{cm} \]

### Step 2: Apply the Mirror Formula for the Concave Mirror
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
For the concave mirror, the focal length \( f \) is negative:
\[ f = -10 \, \text{cm} \]
Substituting the values into the mirror formula:
\[ \frac{1}{-10} = \frac{1}{v_1} + \frac{1}{-15} \]
This simplifies to:
\[ \frac{1}{v_1} = \frac{1}{-10} + \frac{1}{15} \]
Finding a common denominator (30):
\[ \frac{1}{v_1} = \frac{-3 + 2}{30} = \frac{-1}{30} \]
Thus:
\[ v_1 = -30 \, \text{cm} \]

### Step 3: Interpret the Image from the Concave Mirror
The negative value of \( v_1 \) indicates that the image formed by the concave mirror is virtual and located 30 cm in front of the mirror (on the same side as the object).

### Step 4: Calculate the Object Distance for the Convex Mirror
The distance between the two mirrors is 40 cm. The image formed by the concave mirror is 30 cm in front of it, so the distance from the image to the convex mirror is:
\[ d = 40 - 30 = 10 \, \text{cm} \]
For the convex mirror, we take the object distance \( u_2 \) as negative:
\[ u_2 = -10 \, \text{cm} \]

### Step 5: Apply the Mirror Formula for the Convex Mirror
The focal length of the convex mirror is positive:
\[ f = +15 \, \text{cm} \]
Using the mirror formula:
\[ \frac{1}{f} = \frac{1}{v_2} + \frac{1}{u_2} \]
Substituting the values:
\[ \frac{1}{15} = \frac{1}{v_2} + \frac{1}{-10} \]
This simplifies to:
\[ \frac{1}{v_2} = \frac{1}{15} + \frac{1}{10} \]
Finding a common denominator (30):
\[ \frac{1}{v_2} = \frac{2 + 3}{30} = \frac{5}{30} = \frac{1}{6} \]
Thus:
\[ v_2 = +6 \, \text{cm} \]

### Step 6: Interpret the Image from the Convex Mirror
The positive value of \( v_2 \) indicates that the image formed by the convex mirror is virtual and located 6 cm behind the mirror.

### Summary of Results
1. The image formed by the concave mirror (I1) is virtual and located 30 cm in front of the mirror.
2. The image formed by the convex mirror (I2) is virtual and located 6 cm behind the mirror.