Track / Stage 4 / 116 of 340 #376 of 1964

Problem 376

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Find the answer

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 $|A B|$ .

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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 reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.