An object is object at a distance of 25cm from the pole of a convex mirror and a plane mirror is set at a distance 5 cm from convex mirror so that the virtual images formed by the two mirrors do not have any parallax. The focal length of the convex mirror is

Options

A. 37.5cm
B. `-7.5cm `
C. `-37.5cm`
D. `+7.5cm`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the setup
We have a convex mirror and a plane mirror. The object is placed 25 cm from the convex mirror, and the plane mirror is placed 5 cm away from the convex mirror.

### Step 2: Identify the distances
- Distance of the object (O) from the convex mirror (C) = 25 cm (denote this as \( u_{convex} = -25 \) cm, negative as per the sign convention for mirrors).
- Distance between the convex mirror and the plane mirror = 5 cm.
- Therefore, the distance of the object from the plane mirror = \( 25 - 5 = 20 \) cm (denote this as \( u_{plane} = -20 \) cm).

### Step 3: Determine the image distances
For the plane mirror, the image distance (\( v_{plane} \)) is equal to the object distance (in magnitude) since the image formed by a plane mirror is virtual and located behind the mirror:
- Thus, \( v_{plane} = 20 \) cm.

For the convex mirror, the image distance (\( v_{convex} \)) can be calculated as follows:
- Since the plane mirror is 5 cm away from the convex mirror, the image formed by the convex mirror will be at a distance of \( 20 - 5 = 15 \) cm from the convex mirror (denote this as \( v_{convex} = +15 \) cm, positive as per the sign convention).

### Step 4: Apply the mirror formula
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{v} + \frac{1}{u} \]
For the convex mirror:
- \( v = +15 \) cm
- \( u = -25 \) cm

Substituting these values into the mirror formula:
\[ \frac{1}{F} = \frac{1}{15} + \frac{1}{-25} \]

### Step 5: Calculate the focal length
Calculating the right-hand side:
\[ \frac{1}{F} = \frac{1}{15} - \frac{1}{25} \]
Finding a common denominator (which is 75):
\[ \frac{1}{F} = \frac{5}{75} - \frac{3}{75} = \frac{2}{75} \]
Thus,
\[ F = \frac{75}{2} = 37.5 \text{ cm} \]

### Final Answer
The focal length of the convex mirror is \( 37.5 \) cm.

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