Track / Stage 4 / 81 of 340 #341 of 1964

Problem 341

AMC 12 late, AIME early

Geometry Difficulty 4.7 Find the answer

Example 1. Determine what kind of curve the following equations represent:
(1) $\rho=\frac{3}{3-4 \cos \theta}$ ;
(2) $\rho=\frac{6}{4-3 \cos \theta}$ ,
(3) $p=\frac{2}{1+\cos \theta}$ ;
(4) $\rho=\frac{1}{2+3 \sin \theta}$ .

Your answer

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

Official solution

In the table, equation $\mathrm{I}$ , given $e=\frac{4}{3}>1$ , the equation represents a hyperbola;
(2) The original equation can be transformed into $\rho=\frac{\frac{3}{2}}{1-\frac{3}{4} \cos \theta}$ , $e=\frac{3}{4}<1$ , hence it represents a hyperbola.

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