An object is placed in front of a convex mirror at a distance of 50 cm. A plane mirror is introduced covering the lower half of the convex mirror. If the distance between the object and the 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?

As shown in figure, the plane mirror will form erect and virtual image of same size at a distance of 30 cm behind it. So the distance of image formed by plane mirror from convex mirror will be

$\begin{array}{l}\mathrm{PI}=\mathrm{MI}-\mathrm{MP}\\ \mathrm{But}\mathrm{as},\mathrm{MI}-\mathrm{MO}\\ \mathrm{PI}=\mathrm{MO}-\mathrm{MP}=30-20=10\mathrm{cm}\\ \mathrm{Now}\mathrm{as}\mathrm{this}\mathrm{image}\mathrm{coincides}\mathrm{with}\mathrm{the}\mathrm{image}\mathrm{formed}\mathrm{by}\mathrm{convex}\mathrm{mirror},\mathrm{therefore}\mathrm{for}\mathrm{convex}\mathrm{mirror}\\ \mathrm{u}=-50\mathrm{cm};\mathrm{V}=+10\mathrm{cm}\\ \mathrm{So},\frac{1}{+10}+\frac{1}{-50}=\frac{1}{\mathrm{f}}\mathrm{i}.\mathrm{e}.,\mathrm{f}=\frac{50}{4}=12.5\mathrm{cm}\\ \mathrm{So},\mathrm{R}=2\mathrm{f}=2\times 12.5=25\mathrm{cm}\end{array}$