A concave mirror of focal length `10 cm` is placed at a distance of `35 cm` from a wall. How far from the wall should an object be placed to get its image on the wall ?

Explanation

To solve the problem step by step, we will use the mirror formula and the information provided in the question.

### Step 1: Understand the given data
- Focal length of the concave mirror (F) = -10 cm (negative because it's a concave mirror).
- Distance from the mirror to the wall = 35 cm.
- We need to find the object distance (U) from the wall such that the image is formed on the wall.

### Step 2: Set up the coordinate system
- Let the position of the mirror be at the origin (0 cm).
- The wall is located at -35 cm (since we are measuring distances to the left of the mirror).
- The image distance (V) will be -35 cm (the image is formed on the wall).

### Step 3: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Where:
- \( F \) = focal length
- \( V \) = image distance
- \( U \) = object distance

### Step 4: Substitute the known values into the formula
Substituting the values into the mirror formula:
\[ \frac{1}{-10} = \frac{1}{-35} + \frac{1}{U} \]

### Step 5: Rearranging the equation
Rearranging the equation to solve for \( \frac{1}{U} \):
\[ \frac{1}{U} = \frac{1}{-10} - \frac{1}{-35} \]

### Step 6: Finding a common denominator and solving
The common denominator for 10 and 35 is 70. Thus:
\[ \frac{1}{U} = \frac{-7}{70} + \frac{2}{70} = \frac{-7 + 2}{70} = \frac{-5}{70} \]
\[ \frac{1}{U} = \frac{-1}{14} \]

### Step 7: Calculate U
Taking the reciprocal gives:
\[ U = -14 \text{ cm} \]
This means the object is located 14 cm in front of the mirror.

### Step 8: Calculate the distance from the wall
Since the wall is 35 cm away from the mirror, the distance of the object from the wall is:
\[ \text{Distance from the wall} = 35 - 14 = 21 \text{ cm} \]

### Final Answer
The object should be placed **21 cm** from the wall.

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