A child is standing in front of a straight plane mirror. His father is standing behind him, as shown in the fig. The height of the father is double the height of the child. What is the minimum length of the mirror required so that the child can completely see his own image and his father's image in the mirror? Given that the height of the father is 2H.

Be aware that the boy's entire picture requires a minimum mirror length of half of his height. The light beam from the tip of the kid's father must reflect at the topmost point of the mirror and reach the boy's eye in order for the boy to view his father's entire picture. Create a proper ray diagram and use geometry to determine the length of the mirror.
Here, the minimum size of the mirror must be half the kid's height if the boy wishes to view his entire picture, that is, from the bottommost point to the top of his head.

Given that the youngster is H inches tall. The length of the plane mirror for the aforementioned scenario must thus be at least H/2.
The youngster must, however, be able to view his father's entire picture, who is standing behind the boy.
As you can see in the picture, the youngster can see his father's foot but not all of his upper body when using a mirror of length H/2. This implies that the mirror's length must be more than H/2. Let the additional distance be x.

The light beam from the tip of the boy's father's head must reflect and enter his eye as depicted in order for him to see the tip of his head.
The rules of reflection state that $\angle RMQ=\angle PMQ=\theta$.

The length $\mathrm{RP}=\mathrm{H}$ and $\mathrm{QR}=\mathrm{x}$
Therefore, $\mathrm{RQ}=\mathrm{H}-\mathrm{x}$.
In $\Delta RMQ,\tan \theta =\frac{RQ}{2l}=\frac{H-x}{2l}\dots$(i).
In $\Delta M{P}^{\text{'}}M,\tan \theta =\frac{x}{l}\dots \dots$(ii).
From (i) and (ii) we get,
$\frac{H-x}{2l}=\frac{x}{l}$
$\Rightarrow H-x=2x$
$\Rightarrow x=\frac{H}{3}$
Therefore, the mirror's overall length is $x+\frac{H}{2}=\frac{H}{3}+\frac{H}{2}=\frac{5H}{6}$.
This means that the length of the mirror must be more than this if the youngster wishes to view both his complete picture and that of his father $\frac{5H}{6}$, in this instance.

Hence, the correct answer is option B.