A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. The position of image and the magnification respectively are

Options

A. `3.33,5/7`
B. `6.7cm,1.8`
C. 0.15 cm, 1.8
D. `6.7cm,5/9`

Explanation

To solve the problem, we need to find the position of the image and the magnification when a needle of height 4.5 cm is placed 12 cm away from a convex mirror with a focal length of 15 cm.

### Step-by-Step Solution:

1. **Identify the given values**:
- Height of the needle (object height, \( h \)) = 4.5 cm
- Object distance (\( u \)) = -12 cm (the object distance is taken as negative in mirror formula convention)
- Focal length of the convex mirror (\( f \)) = +15 cm (focal length is positive for convex mirrors)

2. **Use the mirror formula**:
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values into the formula:
\[ \frac{1}{15} = \frac{1}{v} + \frac{1}{-12} \]

3. **Rearranging the equation**:
\[ \frac{1}{v} = \frac{1}{15} + \frac{1}{12} \]
To add these fractions, find a common denominator. The least common multiple of 15 and 12 is 60.
\[ \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
Therefore,
\[ \frac{1}{v} = \frac{4}{60} + \frac{5}{60} = \frac{9}{60} \]

4. **Calculate \( v \)**:
Taking the reciprocal gives:
\[ v = \frac{60}{9} = \frac{20}{3} \approx 6.67 \text{ cm} \]

5. **Calculate the magnification (\( m \))**:
The magnification formula for mirrors is given by:
\[ m = -\frac{v}{u} \]
Substituting the values:
\[ m = -\frac{\frac{20}{3}}{-12} = \frac{20}{3 \times 12} = \frac{20}{36} = \frac{5}{9} \]

### Final Results:
- Position of the image (\( v \)) = 6.67 cm (approximately)
- Magnification (\( m \)) = \(\frac{5}{9}\)