1. On a transparent plate, there is an opaque sphere with radius $R$. Above the sphere, on an imaginary line that passes through the center of the sphere and is perpendicular to the plate, there is a point light source, located $a$ ( $a>2$ ) sphere radii away from the plate.

a) Calculate the area of the illuminated surface of the sphere and the area of the shadow of the sphere on the plate.

b) What happens to these areas as $a$ increases indefinitely.
Solution. a) From the similarity of triangles $\triangle C E O$ and $\triangle S C O$ (Figure 3), we obtain

$$
\frac{\overline{O E}}{\overline{C O}}=\frac{\overline{C O}}{\overline{O S}}, \quad \frac{R-\overline{E F}}{r}=\frac{R}{(a-1) R},
$$

i.e.

$$
\overline{E F}=R \frac{a-2}{a-1}
$$

Accordingly, the area of the illuminated part of the sphere is

$$
P_{1}=2 \pi R^{2} \frac{a-2}{a-1}=2 \pi R^{2}\left(1-\frac{1}{a-1}\right)
$$

From the similarity of triangles $\triangle A D S$ and $\triangle S C O$, we obtain:

![](https://cdn.mathpix.com/cropped/2024_06_05_262e89c92fe147e6116eg-6.jpg?height=524&width=511&top_left_y=824&top_left_x=975)

$$
\begin{aligned}
\frac{\overline{A D}}{\overline{S D}}=\frac{\overline{C O}}{\overline{C S}} & \Leftrightarrow \quad \frac{\overline{A D}}{a R}=\frac{R}{\sqrt{(a-1)^{2} R^{2}-R^{2}}} \\
& \Leftrightarrow \quad \overline{A D}=\frac{a R}{\sqrt{a^{2}-2 a}},
\end{aligned}
$$

so the area of the shadow of the sphere on the plate is

$$
P_{2}=\frac{\pi a R^{2}}{a-2}=R^{2}\left(1+\frac{2}{a-2}\right)
$$

b) If $a \rightarrow \infty$, then $P_{1} \rightarrow 2 \pi R^{2}$ and $P_{2} \rightarrow R^{2} \pi$.