A convex mirror of radius of curvature 1.6m has an object placed at a distance of 1m from it. The image is formed at a distance of

Options

A. `8//13m` in fron of the mirror
B. `8//13m` behind the mirror
C. `4//9` m in front of the mirror
D. `4//9m` behind the mirror

Explanation

To solve the problem step by step, we will use the mirror formula and the properties of a convex mirror.

### Step 1: Determine the focal length of the convex mirror
The focal length (F) of a convex mirror is given by the formula:
\[ F = \frac{R}{2} \]
where R is the radius of curvature.

Given:
\[ R = 1.6 \, \text{m} \]
So,
\[ F = \frac{1.6}{2} = 0.8 \, \text{m} \]

### Step 2: Assign the sign convention
In the sign convention for mirrors:
- The focal length (F) of a convex mirror is positive.
- The object distance (U) is negative when the object is in front of the mirror.

Given:
\[ U = -1 \, \text{m} \]

### Step 3: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]

Rearranging the formula to find V:
\[ \frac{1}{V} = \frac{1}{F} - \frac{1}{U} \]

### Step 4: Substitute the values into the formula
Substituting the values of F and U:
\[ \frac{1}{V} = \frac{1}{0.8} - \frac{1}{-1} \]

Calculating \( \frac{1}{0.8} \):
\[ \frac{1}{0.8} = 1.25 \]

So,
\[ \frac{1}{V} = 1.25 + 1 = 2.25 \]

### Step 5: Calculate V
Now, we can find V:
\[ V = \frac{1}{2.25} \]

Calculating \( V \):
\[ V = \frac{1}{2.25} = \frac{4}{9} \, \text{m} \]

### Step 6: Determine the position of the image
Since V is positive, it indicates that the image is formed behind the mirror.

### Final Answer
The image is formed at a distance of \( \frac{4}{9} \, \text{m} \) behind the mirror.

---