Chapter 10, Problem 21RE

A parametric curve of the form

$x=a\cot t+b\cos t,y=a+b\sin t\left(0<t<2\pi \right)$

is called a conchoid of Nicomedes (see the accompanying figure for the case $0<a<b$ ).

(a) Describe how the conchoid $x=\cot t+4\cos t,y=1+4\sin t$ is generated as t varies over the interval $0<t<2\pi .$

(b) Find the horizontal asymptote of the conchoid given in part (a).

(c) For what values of t does the conchoid in part (a) have a horizontal tangent line? A vertical tangent line?

(d) Find a polar equation $r=f\left(\theta \right)$ for the conchoid in part (a), and then find polar equations for the tangent lines to the conchoid at the pole.