Chapter 10.3, Problem 49E

### Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636

Chapter
Section

### Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636
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+2secÎ¸

Calculation:

The polar curve is r=4+2secÎ¸ (1)

The equation for the variable x is given below,

x=rcosÎ¸xr=cosÎ¸secÎ¸=rx

Substitute (rx) for (secÎ¸) in equation (1),

r=4+2(rx)2rx=râˆ’4x2r=1râˆ’4x=2rr(1âˆ’4r)x=2(1âˆ’4r)

Take limit as râ†’Â±âˆž

limrâ†’Â±âˆžx=limrâ†’Â±âˆž2(1âˆ’4r)=2(1âˆ’4limrâ†’Â±âˆž1r)=2(1âˆ’0)=2

Hence, x=2 is a vertical asymptote.

râ†’Â±âˆž4+2secÎ¸â†’Â±âˆžÎ¸â†’(Ï€2)+,(3Ï€2)âˆ’Î¸â†’(Ï€2)âˆ’,(3Ï€2)+

For Î¸ value (Ï€2)

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