Answer

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**Hint:-**The virtual images of both the mirrors should coincide. The position of the image formed by the plane mirror should be equal to the position at which the image formed by the convex mirror also the focal length of the plane mirror is infinite.

Formula used: The formula of the mirror formula is given by,

$\dfrac{1}{u} + \dfrac{1}{v} = \dfrac{1}{f}$

Where u is the object distance from the mirror v is the image distance from the mirror and f is the focal length of the mirror.

**Complete step-by-step solution**

It is given in the problem that a plane mirror is placed in between the convex mirror and the object and we need to find out the value of the focal length of the convex mirror such that the images formed by the plane mirror and the convex mirror coincide with each other.

The object is placed at 20 cm in front of the convex mirror and the plane mirror is 12 cm from the plane mirror.

The focal length of the plane mirror is infinite therefore,

$ \Rightarrow \dfrac{1}{f} = \dfrac{1}{\infty }$

$ \Rightarrow \dfrac{1}{f} = 0$

The mirror formula is,

$ \Rightarrow \dfrac{1}{u} + \dfrac{1}{v} = \dfrac{1}{f}$

$ \Rightarrow \dfrac{1}{u} + \dfrac{1}{v} = 0$

$ \Rightarrow \dfrac{1}{u} = - \dfrac{1}{v}$

$ \Rightarrow v = - u$

The image of the object formed will be virtual.

Therefore the image formed will be at position$v = - u = - 12$.

The distance between the object and the image is equal to,

$ \Rightarrow 12 - \left( { - 12} \right) = 24cm$

Now for the convex mirror the formula will be,

$\dfrac{1}{{u'}} + \dfrac{1}{{v'}} = \dfrac{1}{{f'}}$

Where u’ is the object distance v’ is the image distance and f’ is the focal distance.

The distance of the object is given by $u' = 20cm$ and image distance will be,

$ \Rightarrow v' = - \left( {24 - 20} \right)$

$ \Rightarrow v' = - 4cm$

Now let us calculate the focal length of the convex mirror,

$ \Rightarrow \dfrac{1}{{u'}} + \dfrac{1}{{v'}} = \dfrac{1}{{f'}}$

$ \Rightarrow \dfrac{1}{{20}} - \dfrac{1}{4} = \dfrac{1}{{f'}}$

$ \Rightarrow \dfrac{1}{{f'}} = \dfrac{1}{{20}} - \dfrac{1}{4}$

$ \Rightarrow \dfrac{1}{{f'}} = \dfrac{{1 - 5}}{{20}}$

$ \Rightarrow \dfrac{1}{{f'}} = \dfrac{{ - 4}}{{20}}$

$ \Rightarrow \dfrac{1}{{f'}} = \dfrac{{ - 1}}{5}$

$ \Rightarrow f' = - 5cm$

The focal length of the convex mirror is equal to$f' = - 5cm$. The correct answer for this problem is option B.

**Note:-**The focal length of the plane mirror is infinite which means that the image will be formed at a point which is exactly the same distance as the object is from the plane mirror but on the same side of the object.

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