Olympiad Maths Prep

Track / Stage 4 / 253 of 340 #513 of 2000

Problem 513

AMC 12 late, AIME early
Geometry Difficulty 4.9 Find the answer

Five, find the length of the chord intercepted by the graph x2+y2−2axcos⁡α−2bysin⁡αx^{2}+y^{2}-2 a x \cos \alpha-2 b y \sin \alpha −a2sin⁡2α=0-a^{2} \sin ^{2} \alpha=0 on the yy-axis.

Official solution

Let x=0\mathbf{x}=0, the equation of the circle becomes
y2−2bysin⁡α−a2sin⁡2α=0.  \mathrm{y}^{2}-2 \mathrm{by} \sin \alpha-\mathrm{a}^{2} \sin ^{2} \alpha=0 \text {. }

By Vieta's formulas, we have
y1+y2=2bsin⁡α,y1y2=−a2sin⁡2α, chord length =∣y1−y2∣=(y1+y2)2−4y1y2=4b2sin⁡2α+4a2sin⁡2α=2∣sin⁡α∣a2+b2. \begin{aligned} y_{1} & +y_{2}=2 b \sin \alpha, y_{1} y_{2}=-a^{2} \sin ^{2} \alpha, \\ \text { chord length } & =\left|y_{1}-y_{2}\right|=\sqrt{\left(y_{1}+y_{2}\right)^{2}-4 y_{1} y_{2}} \\ & =\sqrt{4 b^{2} \sin ^{2} \alpha+4 a^{2} \sin ^{2} \alpha} \\ & =2|\sin \alpha| \sqrt{a^{2}+b^{2}} . \end{aligned}

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