(a) With the help of a suitable ray diagram, derive the mirror formula for a concave mirror.

Explanation

Mirror Formula for Concave Mirror : Let, AB be the length of an object placed beyond C in front of a concave mirror. The image A 'B' is real, inverted and between C and F.
Applying sign conventions, we have
object distance `PB=-u`, image distance `PB'=-v`, focal length `PF=-f`
and radius of curvature `PC=-2f`.
In similar `Delta s` ABC and A' B' F
`(AB)/(A'B')=(BC)/(B'C)` ...(i)
and in similar `Delta s` RSF and A' B' F
`(RS)/(A'B')=(SF)/(B'F)" ":' " "RS=AB`
`(AB)/(A'B')=(SF)/(B'F)` ...(ii)
From eq. (i) and (ii), we have
`(BC)/(B'C)=(SF)/(B'F)`
Since, the aperture of the concave mirror is small so the point S and P coincides.
`:. (BC)/(B'C)=(PF)/(B'F)`
`(PB-PC)/(PC-PB')=(PF)/(PB'-PE)`
`(-u+2f)/(-2f+v)=(-f)/(-v+f)`
`uv-uf-2vf+2f^(2)=2f^(2)-fv`
`implies uv=uf+vf`
Dividing both sides by `uvf`, we get
`(uv)/(uvf)=(uf)/(uvf)+(vf)/(uvf)" ":." "1/f=1/v+1/u`.