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a real image one fifth the size of object is formed at a distance of 18 cm from a mirror what is the nature of mirror calculate its focal length

Dear Student,

Real image is always inverted therefore h' (height of image) will be taken negative. Thus,

As the image is real, small in size and focal length is negative so the mirror is concave mirror.

Regards

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M = +1/5  or -1/5         and           v = +18 cm  or -18 cm
So there are 4 possibilities.   But  v positive means behind convex mirror and hence, the image has to be erect.  Hence m is positive only.  There are 3 cases only.

It is a mirror:     m = -v/u               1/f = 1/v + 1/u

case 1)
  m = +1/5  = -v/u      let v = +18 cm   
     u = - 5 v = -90 cm
    1/f = 1/18 - 1/90 = 4/90       =>  f = +22.5 cm    Convex Mirror.

case ii)
   m = +1/5 = -v/u   let  v = -18 cm
       u =  +90 cm      not possible.  as u is always negative.
case iii)
   m = -1/5 = -v/u       let v = +18 cm
        u  = +90 cm      not possible 

case iv)
    m = -1/5    = -v/u        u = 5 v
         v = -18 cm  
          1/f = -1/18 - 1/90 = -6/90
     f = -15 cm           concave mirror.........
HOPE IT WILL HLP YOU............
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