Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

A long horizontal mirror is next to a vertical screen (see figure). Parallel light rays are falling on the mirror at an angle α from the vertical. If a vertical object of height h is kept on the mirror at a distance d>htanα, the length of the shadow of the object on the screen would be.

A. h2
B. htanα
C. h
D. 4h

Answer
VerifiedVerified
143.7k+ views
like imagedislike image
Hint: To solve this problem, we have to understand two important concepts and apply them in our problem: i) Law of reflection: The incident angle is equal to the reflected angle. i=r
ii) The trigonometric ratio, tanθ is the ratio of the opposite side of the angle and its adjacent side. tanθ=OppositeAdjacent

Complete step by step answer:
The law of reflection states that the incident angle is equal to the reflected angle.
The object of height h is placed on the mirror. The shadow of the object falls on the mirror and the same shadow is reflected back on the screen
The corresponding geometry for the same is given in this figure:


Let us understand the formation of the above geometry in detail.
When the object shadow falls on the mirror at the points A and E, it undergoes a reflection on the plane mirror and an image CD is formed on the screen.
At point A, we have a normal N. The angle made by the ray FA is equal to α since the lines EE and AN are parallel lines and AF is the transverse line, which makes the angles AFE and FAN alternate angles which are equal.
At point A, the incident angle is equal to α. Hence, the reflection angle NAD should also be equal to α by the law of the reflection.
Since N is the normal, the angle NAB being equal to 90, the angle CAB = 90α
Since AC is parallel to ED, DEB=CAB=90α corresponding angles.
In the triangle AEF,
tanα=OppositeAdjacent=AEEF=AEh
Thus, AE=htanα
Given that, AE + AB = d
AB=dAE=dhtanα
To calculate the length of the shadow, we need to consider two triangles ABC and EBD to calculate the value of the length CD which is our required value.
In triangle ABC,
tan(90α)=BCAB=BCdhtanα
The trigonometrical identity equal to tan(90θ) is equal to another identity called cotθ, where –
cotθ=1tanθ
Hence,
cotα=BCdhtanα
In triangle EBD,
tan(90α)=BDEB=BDd
The trigonometrical identity equal to tan(90θ) is equal to another identity called cotθ, where –
cotθ=1tanθ
Hence,
cotα=BDd
By equating the values of cotα in the above triangles,
BCdhtanα=BDd
BC×d=BD×(dhtanα)
BC×d=BD×dBD×htanα
BD×htanα=d(BDBC)
The required length, CD=BDBC

Substituting the value of CD and the value of BC=(dhtanα)(cotα), we get –
(CD+BC)htanα=d×CD
(CD+cotα(dhtanα))htanα=d×CD
CD(htanα)+hcotαtanα(dhtanα)=d×CD
Since cotα=1tanα
cotαtanα=1
Thus,
CD(htanα)+h(dhtanα)=d×CD
CD(htanαd)=h(htanαd)
CD=h(htanαd)(htanαd)=h
Hence, the length of the shadow on the screen, CD = h
Hence, the correct option is Option C.

Note: In this answer, the concept of alternate and corresponding angles are directly applied here. Let us briefly understand them.


A and B are two parallel lines and the line C which intercepts them at an angle is called the transverse line. When a transverse line intercepts the parallel lines, we get several sets of angles.
Here, angles a are called corresponding angle pair and angles b are called alternate angle pair. The relation between them is a+b=180
Latest Vedantu courses for you
Grade 11 Science PCM | CBSE | SCHOOL | English
CBSE (2025-26)
calendar iconAcademic year 2025-26
language iconENGLISH
book iconUnlimited access till final school exam
tick
School Full course for CBSE students
PhysicsPhysics
ChemistryChemistry
MathsMaths
₹41,848 per year
EMI starts from ₹3,487.34 per month
Select and buy