Use the mirror equation to deduct that :
(a) an object between `f` and `2f` of a concave mirror produces a real image beyond `2 f`.
(b) a convex mirror always produces a virtual image independent of the location of the object.
( c) the virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole.
(d) an object placed between the pole and focus of a concave mirror produces a virtual and enlarged image.

Explanation

(a) As `DeltasABC` and `A'B'C` are similar
`therefore (AB)/(A'B')=(CB)/(CB') ……..(I)`
Again as `DeltasABP` and `A'B'P` are similar
`(AB)/(A'B')=(PB)/(CB') ……..(ii)`
From `(i)` and `(ii)`
`(AB)/(A'B)=(PB)/(PB') …….(iii)`
Measuring all distance from `P` , we have
`CB=PB-PC`
`CB'=PC-PB'`
`therefore` from `(iii)`
`(PB-PC)/(PC-PB')=(PB)/(PB') ........(iv)`
Using New Cartisian sign conventions,
`PB=-u`
`PC=-R`
`PB'=-v`
We get from `(iv)`
`(-u+R)/(-R+v)=(-u)/(-v)`
or `+uR-uv=uv-vR`
`uR+vR=2uv`
Dividing both sides by `uvR`, we get
`(1)/(v)+(1)/(u)=(2)/(R)`
As `(1)/(v)+(1)/(u)=(2)/(R)=(2)/(2f)=(1)/(f)`
`rArr (1)/(v)+(1)/(u)=(1)/(f)`

(b) According to sign conventions,
`PB'=-u,PB'=+v`
`PB=f,PC=2f`
`DeltasABC` and `A'B'C` are similar
`therefore (AB)/(A'B')=(CB)/(B'C)`
But all distances along the principal axis should be measured from the pole of the mirror.
`(AB)/(A'B')=(PC+PB)/(PC-PB')`
Since the aperture of the mirror is small therefore `MP` can be regarded as a straight line.
`DeltasMPF` and `A'B'F` are similar
`(MP)/(A'B')=(PF)/(B'F)=(PF)/(PF-PB')` or ` (AB)/(A'B')=(f)/(f-v) ........(ii)`
from `(i)` and `(ii)`
`(2f-u)/(2f-v)=(f)/(f-v)`
`2f^(2)-2fv-uf+uv=2f^(2)-vf`
`-fv-uf+uv=0`
`uv=fv+fu`
Dividing by `uvf`, we get
`(1)/(f)=(1)/(u)+(1)/(v)`

(c) `(PB-PC)/(PC-PB')=(PB)/(PB') ........(i)`
Proceeding as above, we get
`(CB)/(CB')=(PB)/(PB') ...........(ii)`
Measuring all distance from `O`, we get
`CB=PC-PB`
`CB'=PC+PB`
From `(i)`
`(PC-PB)/(PC+PB') =(PB)/(PB')`
Using New cartisian sign conventins,
PB=-u,PB=+v`
`PC=-R=(-R+u)/(-R+V)=(-u)/(v)`
or `uR-uv=vR+uv`
`uR+vR=2 uv`
Dividing both sides by `uvR`, we get
`(1)/(v)+(1)/(u)=(2)/(R)=(2)/(2f)=(1)/(f) , (1)/(v)+(1)/(u)=(1)/(f)`