Olympiad Maths Prep

Track / Stage 5 / 149 of 400 #749 of 2000

Problem 749

AIME late
Algebra Difficulty 5.3 Find the answer

2. The curve represented by the equation ρ2sin⁡θ−ρcos⁡2θ−sin⁡θ=0(θ∈(0,π))\rho^{2} \sin \theta-\rho \cos ^{2} \theta-\sin \theta=0(\theta \in(0, \pi)) in the Cartesian coordinate system is \qquad

Official solution

Answer: x=1∪(x2+y2+y=0(y≠0))x=1 \cup\left(x^{2}+y^{2}+y=0(y \neq 0)\right)
Difficulty: Easy
Examination: Polar coordinates, trigonometric functions
Analysis: ρ2sin⁡θ−ρcos⁡2θ−sin⁡θ=0\rho^{2} \sin \theta-\rho \cos ^{2} \theta-\sin \theta=0
⇔(ρsin⁡θ−1)(ρ+sin⁡θ)=0\Leftrightarrow(\rho \sin \theta-1)(\rho+\sin \theta)=0
⇔ρsin⁡θ=1\Leftrightarrow \rho \sin \theta=1 or ρ=−sin⁡θ\rho=-\sin \theta
By θ∈(0,π),ρsin⁡θ=1⇔x=1,ρ=−sin⁡θ⇔x2+y2+y=0(y≠0)\theta \in(0, \pi), \rho \sin \theta=1 \Leftrightarrow x=1, \rho=-\sin \theta \Leftrightarrow x^{2}+y^{2}+y=0(y \neq 0)

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