Question

If a man's face is 25 cm in front of a concave shaving mirror producing erect image of 1.5 times the size of face, focal length of the mirror should be:

• A. 75 cm
• B. 25 cm
• C. -75 cm
• D. 60 cm

Hint:

Using the formula \(m = \frac{f}{f - u}\) directly saves time during exams by avoiding the need to calculate the image distance \(v\) first.
Be extremely careful with sign conventions for \(u\) and \(m\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the object distance (\(u\)) and the magnification (\(m\)) produced by a concave mirror. We need to determine the focal length of the mirror.

Step 2: Key Formula or Approach:
The magnification (\(m\)) for a spherical mirror is related to the focal length (\(f\)) and the object distance (\(u\)) by the formula:
\[ m = \frac{f}{f - u} \]

Step 3: Detailed Explanation:

• According to the Cartesian sign convention, the object is placed in front of the mirror, so:
\[ u = -25\text{ cm} \]

• The shaving mirror produces an erect, virtual image, which means the magnification is positive:
\[ m = +1.5 = \frac{3}{2} \]

• Substitute \(m\) and \(u\) into the magnification formula:
\[ \frac{3}{2} = \frac{f}{f - (-25)} \] \[ \frac{3}{2} = \frac{f}{f + 25} \]

• Cross-multiplying the terms:
\[ 3(f + 25) = 2f \] \[ 3f + 75 = 2f \]

• Solving for \(f\):
\[ 3f - 2f = -75 \] \[ f = -75\text{ cm} \]

• The negative sign mathematically confirms that the mirror is concave, and its focal length is \(75\text{ cm}\).


Step 4: Final Answer:
The focal length of the mirror should be \(-75\text{ cm}\).