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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Show that the polar curve r = 4 + 2 sec θ (called a conchoid) has the line x = 2 as a vertical asymptote by showing that limr→±∞ x = 2. Use this fact to help sketch the conchoid.

To determine

To find: The polar curve r=4+2secθ has the line x=2 as vertical asymptote by showing that limr±y=2 .

Explanation

Given:

The polar curve equation r=4+2secθ

Calculation:

The polar curve is r=4+2secθ (1)

The equation for the variable x is given below,

x=rcosθxr=cosθsecθ=rx

Substitute (rx) for (secθ) in equation (1),

r=4+2(rx)2rx=r4x2r=1r4x=2rr(14r)x=2(14r)

Take limit as r±

limr±x=limr±2(14r)=2(14limr±1r)=2(10)=2

Hence, x=2 is a vertical asymptote.

r±4+2secθ±θ(π2)+,(3π2)θ(π2),(3π2)+

For θ value (π2)

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