By Jaan Kalda (TalTech).
In the figure below, the ellipse is a real image of a circle created by an ideal thin lens whose center is at point O. Geometrically construct the main optical axis of this thin lens, which is known to lie in the plane of this figure. Use the figure in .ggb format, also available here.
For an answer-only submission, provide the factor a from the equation y=ax+b defining the optical axis, using 8 decimal digits. For a complete solution, submit both a GeoGebra file (.ggb) and, if not previously submitted due to technical issues, include the value of a with 8 decimal digits in your email.
Remark 1: You are not supposed to use trial-and-error approach (e.g. looking for such a position of some point by which three points are collinear or two points coincide etc). The optical axis of the lens should come out from the geometrical construction without any room for trials.
Remark 2: Please download and use the Geogebra file with the ellipse and the focus, also available here. With the Geogebra (online app: https://www.geogebra.org/classic), you may use the functionality available through the menu buttons (except the Locus, Polar, Complex numbers, Best Fit Line, functions tools, and tools with numeric input). You are not allowed to make numerical calculations in Geogebra and enter calculated point coordinates. The available tools are shown below.
Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. First hints will appear here on 9th February 2025. After the publication of the first hint, the base score is reduced to 0.9 pts. For full regulations, see the “Participate” tab. The first intermediate results for the Problem 4 will be drawn after 13:00 GMT, 2nd February 2025.