A convex lens of focal length $$20\text{ cm}$$ and a convex mirror are arranged as shown. A point object is located on the principal axis at a distance of $$30\text{ cm}$$ to the left of the lens. If the final image coincides with the object, focal length of the mirror must be
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For the lens, $$u = -30\text{ cm}$$, $$f = 20\text{ cm}$$, so $$\frac{1}{v} = \frac{1}{20} - \frac{1}{30} = \frac{1}{60}$$ and $$v = 60\text{ cm}$$. The mirror is $$50\text{ cm}$$ from the lens, so the rays converge towards a point $$10\text{ cm}$$ behind it. For the image to coincide with the object the rays must retrace their path, so this point is the centre of curvature, $$R = 10\text{ cm}$$ and $$f = \frac{R}{2} = 5\text{ cm}$$.