The image formed by the convex mirrors is 1/n times the object and has a focal length f. What is the difference of object from the mirrors?

The correct option is (n – 1)f.

Step-by-Step Explanation:

A convex mirror always forms a virtual, erect, and diminished image of a real object. According to the Cartesian sign convention:
- The focal length $f$ of a convex mirror is positive ($+f$).
- The magnification $m$ is positive because the image is virtual and erect.

The formula for magnification ($m$) of a spherical mirror in terms of object distance ($u$) and image distance ($v$) is:
$$m=-\frac{v}{u}$$

Given that the image formed is $\frac{1}{n}$ times the size of the object, we have:
$$m=\frac{1}{n}$$

Equating the two magnification relations:
$$-\frac{v}{u}=\frac{1}{n}$$
$$v=-\frac{u}{n}$$

Now, using the mirror formula:
$$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$$

Substitute the expression for $v$ in terms of $u$ into the mirror formula:
$$\frac{1}{-\frac{u}{n}}+\frac{1}{u}=\frac{1}{f}$$
$$-\frac{n}{u}+\frac{1}{u}=\frac{1}{f}$$

Combining the terms on the left-hand side:
$$\frac{1-n}{u}=\frac{1}{f}$$

Solving for the object position $u$:
$$u=(1-n)f=-(n-1)f$$

The negative sign confirms that the object is situated in front of the mirror. Thus, the physical distance of the object from the mirror is:
$$\text{Distance}=\left|u\right|=(n-1)f$$