An object is placed in front of a convex mirror of a radius of curvature `20` cm. Its image is formed `8` cm behind the mirror. The object distance is

Options

A. `20` cm
B. `40` cm
C. `60` cm
D. `80` cm

Explanation

To find the object distance for the given convex mirror scenario, we can follow these steps:

### Step 1: Identify the given values
- Radius of curvature (R) = 20 cm
- Image distance (v) = -8 cm (the negative sign indicates that the image is formed behind the mirror)

### Step 2: Calculate the focal length (f)
The focal length (f) of a convex mirror is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]
Since it's a convex mirror, the focal length is positive:
\[ f = +10 \, \text{cm} \]

### Step 3: Use the mirror formula
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) = focal length
- \( v \) = image distance
- \( u \) = object distance

### Step 4: Substitute the known values into the mirror formula
Substituting \( f = +10 \, \text{cm} \) and \( v = -8 \, \text{cm} \):
\[ \frac{1}{10} = \frac{1}{-8} + \frac{1}{u} \]

### Step 5: Rearrange the equation to solve for \( \frac{1}{u} \)
Rearranging gives:
\[ \frac{1}{u} = \frac{1}{10} + \frac{1}{8} \]

### Step 6: Find a common denominator and calculate
The least common multiple of 10 and 8 is 40. Thus:
\[ \frac{1}{10} = \frac{4}{40} \quad \text{and} \quad \frac{1}{8} = \frac{5}{40} \]
So:
\[ \frac{1}{u} = \frac{4}{40} + \frac{5}{40} = \frac{9}{40} \]

### Step 7: Solve for \( u \)
Taking the reciprocal gives:
\[ u = \frac{40}{9} \approx 4.44 \, \text{cm} \]
Since the object distance is conventionally taken as negative for real objects in front of the mirror:
\[ u = -\frac{40}{9} \approx -4.44 \, \text{cm} \]

### Final Answer
The object distance \( u \) is approximately \( -4.44 \, \text{cm} \).

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