An object is placed in front of a convex mirror at a distance of 50cm. A plane mirror is introduced covering the lower half of the convex mirror. If the distance between the object and plane mirror is 30 cm it is found that there is no parallax between the images formed by the two mirrors. What is the radius of curvature of the convex mirror?

Convex Mirror and Plane Mirror: Radius of Curvature

Given:

• Distance of object from convex mirror: u = -50 cm
• Distance between object and plane mirror: 30 cm
• No parallax between images (images coincide at same position)

Solution:

• The plane mirror forms an image 30 cm behind it.
• Since object is 50 cm from convex mirror and 30 cm from plane mirror, the plane mirror is 20 cm in front of the convex mirror.
• The image from the plane mirror is 30 cm behind the plane mirror, which is 10 cm behind the convex mirror.
• For no parallax, the convex mirror must also form its image at this position: v = +10 cm (behind mirror)

Mirror formula: 1/f = 1/v + 1/u

1/f = 1/(+10) + 1/(-50) 
1/f = 1/10 - 1/50 = (5 - 1)/50 = 4/50 = 2/25 
f = 25/2 = 12.5 cm

Radius of curvature: R = 2f = 2 × 12.5 = 25 cm

Answer: The radius of curvature of the convex mirror is 25 cm