Problem 19P

19. Suppose that we replace the parabolic mirror of Problem 18 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, $C$ is a semicircle with center $O$. A ray of light coming in toward the mirror parallel to the axis along the line $P Q$ will be reflected to the point $R$ on the axis so that $\angle P Q O=\angle O Q R$ (the angle of incidence is equal to the angle of reflection). What happens to the point $R$ as $P$ is taken closer and closer to the axis?


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