Question

A girl with two normal eyes wants to see the full width of her face using a plane mirror. The eye to eye and ear to ear distances of her face are 4 inch and 6 inch respectively. The minimum width of the required mirror is:
A. 1 inch
B. 2 inch
C. 3 inch
D. 4 inch

Answer 1

Hint: We are able to see that object from which light falls on our eyes, either directly or indirectly. The light may be reflected or self-produced (in case of luminous objects). In case we want to view a full object having some dimensions, the light from the left and right most corners and from the bottom and top most corner must fall on the eyes.

Complete answer:

Here – A and D are ears; B and C are eyes.

In the figure, we can see that for viewing full face in mirror$M_1 \ - \ M_2$, she need to have part ‘FE’ (Parallel sides) of the mirror for minimum part of the mirror for the rays to be reflected and hence all the face will be covered. Now, in the question, given ear separation = 6 inch and eye to eye distance = 4 inch.
So,$AD = 6 \ inch \ and \ BC = 4\ inch$, where ‘o’ is the centre.
Minimum width needed =$GH = FE$
Now, $AG = \dfrac{AC}{2} = DH = \dfrac{BD}{2}$
Now, note that $AC = Ao + oC = \dfrac{AD}{2}+ \dfrac{BC}{2} = \dfrac 62 + \dfrac 42 =5\ inch = BD$
Also $GH = AD - (AG + HD) = AD - (\dfrac{AC}{2}+\dfrac{BD}{2}) = 6 - (\dfrac52 + \dfrac 52) = 6-5 =1\ inch = FE$

So, the correct answer is “Option A”.

Note:
These types of questions can be easily solved by using symmetry rather than equations. The girl just needs to see the left ear from the right eye and the right ear from the left eye to see the full wide face. One can think that she just has to see the left ear from the left eye and the right ear from the right eye as they are the closest to each other. But in that case she has to see away from the centre of the mirror and hence the width won’t be minimum but more because when she will see in right, there has to be a mirror too, which means that the mirror should also be present in the right part, thereby increasing the width of the mirror.