Track / Stage 5 / 35 of 400 #635 of 1964

Problem 635

AIME late

Geometry Difficulty 5.1 Find the answer

We have two perpendicular lines - like the axes of an ellipse - and a tangent $(t)$ with its point of tangency $(T)$ . Construct the endpoints of the ellipse's axes $(A, B, C, D)$ ! How many solutions are there?

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Let the tangent touch one of the axes at $P$ ; draw a circle over the diameter $\overline{O P}$ and project $T$ perpendicularly onto this circle in the direction of $|O P|$ . $\overline{O T_{1}}$ is the radius of the affine circle from which we obtain $\overline{A B}$ and $\overline{C D}$ as shown in the figure. There is one solution.

Sándor Gyula (Kölcsey Ferencg. VIII. o. Budapest)

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.