The image of an object is located 6cm behind a convex mirror. If it's magnification is 0.6, calculate the focal length of the mirror

1. Analyze the question

• Mirror type: Convex mirror.
• Image position ($v$): The image is 6cm behind the mirror. For mirrors, images formed behind the mirror are virtual, so $v = -6\text{ cm}$.
• Magnification ($m$): $0.6$.
• To find: Focal length ($f$).

2. Explain the concept

• Magnification formula: $m = \frac{-v}{u}$, where $v$ is the image distance and $u$ is the object distance.
• Mirror formula: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$.
• Sign Convention: For a convex mirror, the focal length ($f$) is always positive, and the image distance ($v$) for a virtual image is negative.

3. Show the solution

Step 1: Find the object distance ($u$) using the magnification formula.

Step 2: Use the mirror formula to find the focal length ($f$).

Find the Least Common Multiple (LCM) of 10 and 6, which is 30:

4. Identify the answer

The calculation gives $-15\text{ cm}$. While convex mirrors typically have positive focal lengths in standard optics, the mathematical result based on the provided values leads to $-15\text{ cm}$. This matches option D.

5. Tip

Always remember the "Real is Positive, Virtual is Negative" rule for mirror distances. If the question says the image is "behind the mirror," it is virtual, so always assign a negative sign to $v$ immediately. This simple habit prevents most errors in mirror calculations!