The radius of concave surface is 20 cm and radius of convex surface is 10 cm. The refractive index of material of lens is 1.5. If image of point object P coincides with it. Find the distance (in cm) between object and lens.

$\begin{array}{l}\mathrm{For} \mathrm{pure} \mathrm{lens}, \\ \frac{1}{{\mathrm{f}}_1}=(1.5-1)(-\frac{1}{20}+\frac{1}{10})=\frac{1}{2}\left(\frac{-1+2}{20}\right)=\frac{1}{40}\\ \therefore {\mathrm{P}}_1=\frac{1}{40}\\ \mathrm{and} \\ \mathrm{For} \mathrm{pure} \mathrm{mirror}, \\ {\mathrm{P}}_{\mathrm{m}}=-\frac{1}{{\mathrm{f}}_{\mathrm{m}}}=\frac{1}{5}\\ \mathrm{Combination} \mathrm{of} \mathrm{lens} \mathrm{and} \mathrm{mirror}\\ \therefore \mathrm{P}={\mathrm{P}}_{\mathrm{m}} + 2{\mathrm{P}}_1=\frac{1}{5}+\frac{1}{20}=\frac{1+4}{20} = \frac{1}{4}\\ \Rightarrow \mathrm{P}=\frac{-1}{\mathrm{f}}=\frac{1}{4}\\ \mathrm{So}, \mathrm{equivalent} \mathrm{mirror} \mathrm{is} \mathrm{concave} \mathrm{mirror} \mathrm{of} \mathrm{focal} \mathrm{length} 4 \mathrm{cm}.\\ \mathrm{Hence}, \mathrm{object} \mathrm{should} \mathrm{be} \mathrm{at} \mathrm{the} \mathrm{centre} \mathrm{of} \mathrm{curvature} \mathrm{of} \mathrm{equivalent} \mathrm{concave} \mathrm{mirror} \mathrm{i}.\mathrm{e} 8 \mathrm{cm}.\\ \end{array}$