Question

If the distance between object and its two times magnified virtual image produced by a curved mirror is 15 cm, the focal length of the mirror must be :

• A. 15 cm
• B. -12 cm
• C. -10 cm
• D. 10/3 cm

Answer 1

The Correct Option is C

Given:  
- Magnification, \( m = +2 \) (since the image is virtual and magnified).  
- Distance between the object and the image, \( |u - v| = 15 \, \text{cm} \).

For a mirror, the magnification \( m \) is given by:  

\(m = -\frac{v}{u}\)

Since \( m = +2 \):  

\(-\frac{v}{u} = 2 \implies v = -2u\)

Step 1. Set up the distance equation:  

  \(|u - v| = 15\)
 
  Substitute \( v = -2u \):  
 
 \( |u - (-2u)| = 15\)
 
 \( |3u| = 15\)
  
 \( u = 5 \, \text{cm}\)
 
Step 2. Calculate \( v \):  
 
  \(v = -2u = -2 \times 5 = -10 \, \text{cm}\)
  
Step 3. Use the mirror formula:  
  The mirror formula is:  \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)

  Substitute \( u = 5 \, \text{cm} \) and \( v = -10 \, \text{cm} \):  

  \(\frac{1}{f} = \frac{1}{-10} + \frac{1}{5} = -\frac{1}{10} + \frac{2}{10} = \frac{1}{10}\)

  \(f = 10 \, \text{cm}\)
 

Thus, the focal length of the mirror is \(-10 \, \text{cm}\).

The Correct Answer is : -10 cm