An object is placed on principal axis of concave mirror at distance of 1.5 f Image will be at

Options

A. -3f
B. 1.5f
C. -1.5f
D. 3f

Answer

To solve the problem of finding the image position when an object is placed at a distance of 1.5f from a concave mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Let the focal length of the concave mirror be denoted as \( f \).
- The object distance \( u \) is given as \( -1.5f \) (the negative sign indicates that the object is in front of the mirror).

2. **Use the Mirror Formula:**
- The mirror formula is given by:
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
- Here, \( v \) is the image distance, and \( u \) is the object distance.

3. **Substitute the Values into the Mirror Formula:**
- Substitute \( u = -1.5f \) into the mirror formula:
\[ \frac{1}{v} + \frac{1}{-1.5f} = \frac{1}{f} \]

4. **Rearranging the Equation:**
- Rearranging gives:
\[ \frac{1}{v} - \frac{1}{1.5f} = \frac{1}{f} \]

5. **Finding a Common Denominator:**
- The common denominator for the fractions is \( 1.5f \):
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{1.5f} \]
- Convert \( \frac{1}{f} \) to have a denominator of \( 1.5f \):
\[ \frac{1}{f} = \frac{1.5}{1.5f} \]
- Thus, we can write:
\[ \frac{1}{v} = \frac{1.5}{1.5f} + \frac{1}{1.5f} = \frac{1.5 + 1}{1.5f} = \frac{2.5}{1.5f} \]

6. **Calculate \( v \):**
- Inverting both sides gives:
\[ v = \frac{1.5f}{2.5} = \frac{1.5}{2.5}f = \frac{3}{5}f \]
- Since \( v \) is negative for a concave mirror (indicating that the image is formed on the same side as the object), we have:
\[ v = -3f \]

### Final Answer:
The image will be formed at a distance of \( -3f \) from the mirror.

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