Question

The image formed by a convex mirror is only one-third of the size of the object. If the focal length of the mirror is 12 cm the
image formed will be A .8 cm behind the mirror B. 10 cm behind the mirror C. 8 cm in front of the mirror D. 10 cm in front of
the mirror​

Answer 1

Answer :-

(A) 8 cm behind the mirror

Explanation :-

Image formed by a convex mirror is 1/3 of the object

So Magnification (m) = 1/3

Also Magnification (m) = - v/u

⇒ 1/3 = - v/u

⇒ 3 = - u/v

⇒ 3v = - u

⇒ - 3v = u

⇒ u = - 3v

Focal length (f) = 12 cm

Image distance = v

Object distance (u) = - 3v

We know that

1/f = (1/v) + (1/u)

⇒ 1/12 = (1/v) + (1/-3v)

⇒ 1/12 = (1/v) + (-1/3v)

⇒ 1/12 = (3 - 1)/3v

⇒ 1/12 = 2/3v

⇒ 3v/12 = 2

⇒ v/4 = 2

⇒ v = 2(4)

⇒ v = 8

Since object distance is positive image is formed behind the mirror at a distance of 8 cm.

Answer 2

Given

image formed by a convex mirror is only

one-third of the size of the object.

image formed by a convex mirror is only

one-third of the size of the object. focal

length of the mirror is 12 cm

To find

Image distance (v)

Explanation:

Given,

for a convex mirror,

Focal length=f=12cm

Image formed is 1/3rd the size of the object

hi=ho/3

hi/ho=1/3

but hi/ho is equal to magnification which is also equal to -v/u

therefore

-v/u=1/3

u=-3v..............(1)

Using the formula

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

$= > \dfrac{1}{12} = \dfrac{1}{v} + \dfrac{1}{u } \\ \\ = > \dfrac{1}{12} = \dfrac{1}{v} - \dfrac{1}{3v} \\ \\ = > \dfrac{1}{12 } = \dfrac{3 - 1}{3v} \\ \\ = > \dfrac{1}{12} = \dfrac{2}{3v} \\ \\ = > 3v = 24 \\ \\ \boxed{ \sf \: v = 8cm}$

Therefore

the image will be formed 8cm behind the mirror