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, we need to determine how far from the wall an object should be placed so that its image is formed on the wall using a concave mirror with a given focal length.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Focal length of the concave mirror, \( f = -10 \) cm (negative because it is a concave mirror).
- Distance from the mirror to the wall, \( d_{mw} = 35 \) cm.

2. **Determine the Image Distance:**
- The image is formed on the wall, which is located at a distance of 35 cm from the mirror. Therefore, the image distance \( v = -35 \) cm (negative because the image is real and formed on the same side as the object).

3. **Use the Mirror Formula:**
- The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Plugging in the values we have:
\[ \frac{1}{-10} = \frac{1}{-35} + \frac{1}{u} \]

4. **Rearranging the Equation:**
- Rearranging the equation to find \( \frac{1}{u} \):
\[ \frac{1}{u} = \frac{1}{-10} - \frac{1}{-35} \]

5. **Finding a Common Denominator:**
- The common denominator of 10 and 35 is 70. Therefore:
\[ \frac{1}{u} = \frac{-7}{70} + \frac{2}{70} = \frac{-5}{70} \]

6. **Calculating \( u \):**
- Now, solving for \( u \):
\[ u = \frac{70}{-5} = -14 \text{ cm} \]
- The negative sign indicates that the object is located 14 cm in front of the mirror.

7. **Calculating the Distance from the Wall:**
- Since the wall is 35 cm away from the mirror, the distance from the wall to the object \( d \) can be calculated as:
\[ d = d_{mw} - |u| = 35 - 14 = 21 \text{ cm} \]

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

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