A concave spherical mirror with a focal length of 11 cm faces a plane mirror wit

Question

A concave spherical mirror with a focal length of 11 cm faces a plane mirror with the optical axis of the spherical mirror perpendicular to the plane mirror. A small object is placed at point P on the optical axis, 12 cm from the plane mirror and 31 cm from the vertex of the spherical mirror. Find the distance from the plane mirror to the three nearest images. (Enter your answers from smallest to largest.)

Please show work! This is terribly confusing to me.

Explanation / Answer

Firstly lets discuss the first image of object itself formed by plane and spherical mirror.

by plane mirror:

image will be at same distance as object.

object is at 12 cm from the plane mirror.

i = 12 cm

by spherical mirror:

using mirror equation, 1/f = 1/i + 1/o

for concave f= 11 cm and o = 31 cm

1/11 = 1/i + 1/31

i = 17.05 cm from spherical mirror.

distance from plane mirror = 12 + 31 - 17.05 = 25.95 cm

now these image will work as object for mirror and secondly images will be formed of these
images.

now, obect distance for plane mirror = 25.95 cm

so image distance = 25.95 cm (back to the plane mirror)

for spherical mirror:

object distance = 12 + 31 + 12 = 55 cm

1/11 = 1/55 + 1/i

i = 13.75 cm

distance from plane mirror = 12 + 31 - 13.75 = 29.25 cm

First nearest image = 12 cm back to the mirror

second = 25.95 cm (back and front both sides , two different images)

third = also at 25.95 cm

but third nearest distance = 29.25 cm

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