Track / Stage 4 / 104 of 340 #364 of 1964

Problem 364

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Find the answer

3. Let $A$ and $B$ be two moving points on the ellipse $x^{2}+3 y^{2}=1$ , and $O A \perp O B$ (where $O$ is the origin). Find the maximum and minimum values of $|A B|$ .

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

(Tip: Establish a polar coordinate system with $O$ as the pole and the positive half-axis of the $x$ -axis as the polar axis. Then the equation of the ellipse is $\rho^{2}\left(\cos ^{2} \theta+3 \sin ^{2} \theta\right)=1, \theta \in$ $[0,2 \pi)$ . Let $A\left(\rho_{1}, \alpha\right), B\left(\rho_{2}, \alpha+\frac{\pi}{2}\right)$ , then $|A B|^{2}=\rho_{1}^{2}+$ $\rho_{2}^{2}=\frac{4}{3+\sin ^{2} 2 \alpha} .|A B|$ has a maximum value of $\frac{2}{3}$ and a minimum value of 1. .)

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