A convex lens of focal length $$10\text{ cm}$$ and a convex mirror of focal length $$f_m$$ are mounted coaxially (above, right). The image of a point object located on the axis $$15\text{ cm}$$ to the left of the lens coincides with the object itself. Focal length $$f_m$$ of the mirror is
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For the lens, $$u = -15\text{ cm}$$ and $$f = 10\text{ cm}$$, so $$\frac{1}{v} = \frac{1}{10} - \frac{1}{15} = \frac{1}{30}$$ and $$v = 30\text{ cm}$$. The mirror is $$10\text{ cm}$$ from the lens, so the rays converge towards a point $$20\text{ cm}$$ behind the mirror. The final image coincides with the object only if the rays retrace their path, i.e. strike the mirror normally, so this point is the centre of curvature. Hence $$R = 20\text{ cm}$$ and $$f_m = \frac{R}{2} = 10\text{ cm}$$.