Radius of curvature of a concave mirror is 100 cm. An object is kept at a distance of 25 cm. The position and nature of the image of that object is

Options

A. 50 cm behind mirror, virtual
B. 55 cm behind the mirror, virtual
C. 50 cm behind the mirror, real
D. 55 cm behind the mirror, real

Explanation

To solve the problem step by step, we will use the mirror formula and the relationship between the radius of curvature and the focal length of a concave mirror.

### Step 1: Determine the Focal Length
The radius of curvature (R) of the concave mirror is given as 100 cm. The relationship between the radius of curvature and the focal length (F) is given by the formula:

\[ F = \frac{R}{2} \]

Substituting the value of R:

\[ F = \frac{100 \, \text{cm}}{2} = 50 \, \text{cm} \]

Since it is a concave mirror, the focal length is taken as negative:

\[ F = -50 \, \text{cm} \]

### Step 2: Identify the Object Distance
The object distance (U) is given as 25 cm. In mirror conventions, the object distance is taken as negative when it is in front of the mirror:

\[ U = -25 \, \text{cm} \]

### Step 3: Use the Mirror Formula
The mirror formula is given by:

\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]

Substituting the values of F and U into the mirror formula:

\[ \frac{1}{-50} = \frac{1}{V} + \frac{1}{-25} \]

### Step 4: Simplify the Equation
Rearranging the equation gives:

\[ \frac{1}{V} = \frac{1}{-50} + \frac{1}{25} \]

Finding a common denominator (which is 50):

\[ \frac{1}{V} = \frac{-1}{50} + \frac{2}{50} = \frac{1}{50} \]

### Step 5: Solve for V
Taking the reciprocal to find V:

\[ V = 50 \, \text{cm} \]

### Step 6: Determine the Nature of the Image
The positive value of V indicates that the image is formed on the same side as the object (the right side of the mirror). Since the object is placed between the focal point (F) and the mirror, the image formed will be virtual and erect.

### Conclusion
The position of the image is at 50 cm from the mirror, and the nature of the image is virtual and erect.

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