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

The problem involves finding the focal length of a curved mirror when the distance between an object and its two times magnified virtual image is given. Let us solve this step-by-step. 

1. Given that the image is virtual and magnified, the mirror in use is a concave mirror. In a concave mirror, magnification (\(m\)) is given by: \(m = -\frac{v}{u}\), where \(v\) is the image distance, and \(u\) is the object distance.
2. We are told that the image is two times magnified, so \(m = 2\). Thus, we get: \(2 = -\frac{v}{u}\) leading to \(v = -2u\).
3. The distance between the object and the image is given as 15 cm. Since this is the absolute value of \(v - u\), we have: \(|-2u - u| = 15\) simplifying to \(|3u| = 15\).
4. This gives us \(u = -5\) cm because the object distance for a virtual image (formed by a mirror) is considered negative.
5. Substituting \(u = -5\) cm in \(v = -2u\), we get: \(v = 10\) cm.
6. Now, using the mirror formula: \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\), substitute the values: \(\frac{1}{f} = \frac{1}{10} + \frac{1}{-5}\).
7. Solving the above equation gives: \(\frac{1}{f} = \frac{1}{10} - \frac{2}{10} = \frac{-1}{10}\). Therefore, the focal length \(f = -10\) cm.

Hence, the focal length of the mirror is -10 cm. The correct answer is -10 cm, which matches with the provided correct answer option.

Answer 2

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