290. On the surface of the Earth, all possible points are taken, the geographical latitude of which is equal to their longitude. Find the geometric locus of the projections of these points onto the equatorial plane.
290. Let $O$ be the center of the Earth, $A$ be the point on the equator corresponding to the zero meridian, $M$ be a point on the surface of the Earth with longitude and latitude equal to $\varphi$, and $N$ be the projection of $M$ onto the equatorial plane. By introducing a rectangular coordinate system in the equatorial plane, taking the line $O \AA$ as the $x$-axis and the origin at point $O$, we obtain that $N$ has coordinates: $x = R \cos^2 \varphi, y = R \cos \varphi \sin \varphi$, where $R$ is the radius of the Earth. It is easy to verify that the coordinates of points

![](https://cdn.mathpix.com/cropped/2024_05_21_9b03e22d2b8ca67aa653g-139.jpg?height=357&width=465&top_left_y=1199&top_left_x=165)

Fig. 56.

$A$

and $N$ satisfy the equation

$$
\left(x - \frac{R}{2}\right)^2 + y^2 = \frac{R^2}{4}
$$

i.e., the desired set is a circle with center $\left(\frac{R}{2}, 0\right)$ and radius $\frac{R}{2}$.