Question

A concave mirror has a focal length of 36.0 cm. The distance between an object and its image is 53.7 cm. Find (a) the object and (b) image distances, assuming that the object lies beyond the center of curvature and (c) the object and (d) image distances, assuming that the object lies between the focal point and the mirror.

Answer 1

Focal length of the mirror (f) = 36 cm
Image distance from the pole = d_i
object distance from the pole = d_o
When the object is beyond the center of curvature  
By applying mirror formula
1/d_o + 1/d_i = 1/f
1/d_o + 1/d_i = 1/36 ------------(1)
We know that the distance between image and object
d_o - d_i = 53.7 cm
d_o = (d_i+53.7)
Putting the value in above equation
1/(d_i+53.7) +(1/d_i) = 1/36
(b)d_i = 54.06 cm
(a)d_o = d_i + 53.7 = 107.76 cm
When the object is between focal point and mirror
From the figure
distance between the image and object = 53.7 cm
d_o + d_i = 53.7 cm
d_o = 53.7 - d_i
now from mirror formula
1/d_o +(1/-di) = 1/f
1/(53.7-d_i) -(1/d_i) = 1/36
On solving it we get
d_i = 35.76 cm
d_o = 17.94 cm
Therfore
(c) d_o = 17.94 cm
(d) d_i = 35.76 cm