## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 11: Parametric Equations and Polar Coordinates - Section 11.3 - Polar Coordinates - Exercises 11.3 - Page 663: 38

#### Answer

A hyperbola whose center is at the origin (0,0) with transverse axis $y=x$ and conjugate axis $y=-x$.

#### Work Step by Step

The conversion of polar coordinates and Cartesian coordinates are described as follows: 1. $r^2=x^2+y^2$ and $r=\sqrt {x^2+y^2}$ 2. $\tan \theta =\dfrac{y}{x} \implies \theta =\tan^{-1} (\dfrac{y}{x})$ 3. $x=r \cos \theta$ and 4. $y=r \sin \theta$ Given: $r^2 \sin 2 \theta=2$ This can be rewritten as: $r^2 (2 \sin \theta \cos \theta)=2$ Now, the cartesian equation is $xy=1$ and $y=\dfrac{1}{x}$ Hence, this shows the hyperbola whose center is at the origin (0,0) with transverse axis $y=x$ and conjugate axis $y=-x$.

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