Question

Real image 1/5th size of object is formed at a distance of 18 cm from the mirror. what is the nature of mirror ? calculate its focal length .

Answer 1

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

Answer 2

I hope this will help you.