An object is held at 30 cm in front of a convex mirror of a focal length 15 cm. At what distance from the convex mirror should a plane mirror be held so that images in the two mirrors coincide with each other?

Step 1, Given data and related image

The mirror is a convex mirror

Object distance, u = -30 cm

focal length f = 15 cm

Step 2,

From the mirror formula

We know,

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

Putting all the values

$\frac{1}{v}-\frac{1}{30}=\frac{1}{15}$

$\frac{1}{v}=\frac{1}{30}+\frac{1}{15}$

$\frac{1}{v}=\frac{3}{30}$

After solving

v = 10 cm

Now we can say that the separation between the object and image is 30+10=40 cm.

Now,

For the image to be formed at the same position by a plane mirror the plane mirror must be kept at half of the distance between the object and image.

Half the distance between the object and the image is 20 cm so the plane mirror must be placed at 10 cm from the mirror.