Let us find #(dy)/(dx)# in terms of #r# and #theta# using #x=rcostheta# and #y=rsintheta#.
As such #(dx)/(d theta)=(dr)/(d theta)costheta-rsintheta#
and #(dy)/(d theta)=(dr)/(d theta)sintheta+rcostheta#
i.e. #(dy)/(dx)=((dr)/(d theta)sintheta+rcostheta)/((dr)/(d theta)costheta-rsintheta)#
As #r=3cos2thetasectheta#
#(dr)/(d theta)=-6sin2thetasectheta+3cos2thetasecthetatantheta#
and #(dy)/(dx)=0#, when
#sintheta(3cos2thetasecthetatantheta-6sin2thetasectheta)+rcostheta=0#
or #3cos2thetatan^2theta-12sin^2theta+rcostheta=0#
or #r=(12sin^2theta-3cos2thetatan^2theta)/(costheta)#
or #r=12sin^2thetasectheta-3cos2thetasecthetatan^2theta#
or #r=12sin^2thetasectheta-rtan^2theta#
i.e. #r=(12sin^2thetasectheta)/(1+tan^2theta)=(2tantheta)/(1+tan^2theta)*6sintheta#
= #6sin2thetasintheta#
or #3cos2thetasectheta=6sin2thetasintheta#
or #cos2theta=sin^2 2theta#
or #cos2theta=1-cos^2 2theta#
or #cos^2 2theta+cos2theta-1=0#
or #cos2theta=(-1+sqrt5)/2=0.618#
Observe that we cannot have #cos2theta=(-1-sqrt5)/#
or #2theta=+-51.83^@#
or #theta=+-25.92^@#
i.e. at #y=rsintheta=+-3cos51.83^@sec25.92^@sin25.92^@#
= #3xx0.618xx1.112xx0.437=0.9#
The graph of curve is shown below:
graph{(x^3+xy^2-3x^2+3y^2)(y^2-0.81)=0 [-5, 5, -2.5, 2.5]}
