Size of image of an object by a mirror having a focal length of `20 cm` is observed to be reduced to `1/3 rd` of the size ? At what distance the object has been placed from the mirror ? What is the nature of the image and the mirror ?

Explanation

Here, considering (the case for both type of possible spherical mirrors)
For concave mirror
Focal length `f=-20cm`
Magnification, `m=-(1)/(3)`
Since, magnification,` m=-(v)/(u)`
`:.` Magnification, `m=-(1)/(3)=-(v)/(u)`
`v=(u)/(3)`
Using mirror formula, we have
`(1)/(v)+(1)/(u)=(1)/(f)implies(1)/(f)=(1)/(u)+(3)/(u)=(4)/(u)`
`implies u=4f=4(-20)=-80cm`
The object should be placed at a distance 80 cm from the concave mirror.
For convex mirror,
Focal length, ` f=+20 cm`
Magnification, `m=+(1)/(3)`
Since, magnification,` m=-(v)/(u)`
`:.` Magnification, `m=(1)/(3)=-(v)/(u)`
`v=-(u)/(3)`
Using mirror formula, we have
`(1)/(v)+(1)/(u)=(1)/(f)`
`(1)/(f)=(-3)/(u)+(1)/(u)=(-2)/(u)`
`implies u=-2f=-2(20)=-40cm`
The object should be placed at a distance 40 cm from the convex mirror to form virtual, erect and diminished image.