The distance between a real object and its image in a convex mirror of focal length 12 cm is 32 cm. Find the size of image if the object size is 1 cm.
Step 1: Given that:
Distance between object and image = 32cm
Focal length(f) of the convex mirror = +12cm
If u be the object distance and v be the image distance from the convex mirror then,
u+v=32cm
v=32−u
Size of the object
(ho) = 1cm
Note: The focal length of a convex mirror is taken as positive as it lies right to the pole of the mirror.)
Step 2: Formula used:
The mirror formula is;
1v+1u=1f
Where u = object distance, v = image distance and f = focal length of the mirror
Magnification is given by;
m=hiho=−vu
Step 3: Calculation of the object distance:
Using the mirror formula;
132−u+1−u=1+12cm
132−u−1u=112u−(32−u)u(32−u)=112
2u−3232u−u2=112
2u×12−32×12=32u−u2
⇒u2−32u+24u−32×12=0
⇒u2−8u−384=0
⇒(u2−24u)+(16u−384)=0
⇒u(u−24)+16(u−24)=0
⇒(u−24)(u+16)=0
When,(u−24)≠0
(u+16)=0
u=−16cm
Thus,
v=32−(−16cm)
v=48cm
Andif,(u+16)≠0
u=24cm
Thus,
v=32−24cm
v=8cm
Step 4: Calculation of the size of the image:
For u= 24cm
hi1cm=−48cm−16cm
hi1cm=31
hi=3cm
And
for, u= 24cm
hi1cm=−8cm−24cm
hi=13cm
Thus,
the size of the image is 13cm .