How do you approach and solve this problem?

A curve called a witch of Maria Agnesi consists of all possible positions of the point P in the figure. Show that parametric equations for this curve can be written as
#x=2acottheta#
#y=2asin^2theta#
Sketch the curve.
Stewart's Calculus Concepts and Contexts 4E

1 Answer
Mar 14, 2018

Please see below.

Explanation:

As point #A# moves anticlockwise starting from point #O#, we observe that angle #theta# changes from #0# to #2pi#. Further, coordinates of #P(x,y)# depend on one - #a# the radius of the circle, which is a constant and two - #theta#, which we have seen as varying from #0# to #2pi#. Hence, we ought to work out for #x# and #y#, using these.

For this purpose and to explain it better, I intend to draw a perpendicular from #PN# to #x#-axis. and also name the diameter as #OL# and perpendicular #AM# on #x#-axis#. Also observe that

enter image source here

#m/_OLA=m/_LCO=theta#

Now #x=ON=NCxxcottheta=2acottheta#

and #y=PN=AM=OAsintheta#

but in #DeltaOLA#, #(OA)/(OL)=sin/_OLA=sintheta#

Therefore #y=OAsin^2theta=2asin^2theta#

Hence, parametric form of all possible positions of the point #P# is

#(2acottheta,2asin^2theta)#

We can also find locus in rectangular coordinates too by eliminating #theta# from these equations.

Observe that #x^2=4a^2cot^2theta#

= #4a^2(csc^2theta-1)#

= #4a^2(1/sin^2theta-1)#

i.e. #x^2=4a^2((2a)/y-1)#

and if #a=5# this becomes #x^2=1000/y-100# and graph (includes the circle too) appears as

graph{(1000/y-100-x^2)(x^2+y^2-10y)=0 [-22.5, 22.5, -7.71, 14.79]}