A concave mirror and a plane mirror are arranged facing each other and an object $AB$ is placed between them as shown in the figure. What is the distance between the object and its inverted, virtual image formed in the plane mirror?

Step 1: Ray diagram

Step 2: Finding the image distance:

Let the distance of the object $AB$from the concave mirror be $u$, image distance be $v$ and focal length be $f$

Using the mirror formula, $\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$

$$\begin{gathered} \frac{1}{f}-\frac{1}{u}=\frac{1}{v} \\ \frac{u-f}{uf}=\frac{1}{v} \end{gathered}$$

$v=\frac{uf}{u-f}$ units

An inverted, magnified and the real image is formed at $\frac{uf}{u-f}$units from the concave mirror which acts as an object to the plane mirror.

Step 3: Finding the distance between the object and the image:

Let the distance of the object $AB$ from the plane mirror be $u_1$,

$\Rightarrow$The inverted virtual image is formed at a distance of $u_1+u-\frac{uf}{u-f}$ units behind the plane mirror.

Let $u_1+u-\frac{uf}{u-f}=x$

$\Rightarrow$The distance between the object and its virtual inverted image is $x+u_1$units.

Hence, the distance between the object and its inverted, virtual image formed in the plane mirror is $x+u_1$units.