To convert #2y=-x^2+3x#
Use #x=r*cos theta# and #y=r*sin theta#
Let's do it
#2y=-x^2+3x#
#2(r*sin theta)=-(r*cos theta)^2+3(r*cos theta)#
#2*r*sin theta=-r^2*cos^2 theta+3*r*cos theta#
divide both sides of the equation by #r#
#(2*r*sin theta)/r=(-r^2*cos^2 theta)/r+(3*r*cos theta)/r#
#(2*cancelr*sin theta)/cancelr=(-cancelr^2*cos^2 theta)/cancelr+(3*r*cos theta)/cancelr#
#2*sin theta=-r*cos^2 theta+3*cos theta#
Transposition
#r*cos^2 theta=3*cos theta-2*sin theta#
divide both sides by #cos^2 theta#
#(r*cos^2 theta)/cos^2 theta=(3*cos theta-2*sin theta)/cos^2 theta#
#(r*cancelcos^2 theta)/cancel(cos^2 theta)=(3*cos theta-2*sin theta)/cos^2 theta#
#color(red)(r=(3*cos theta-2*sin theta)/cos^2 theta)#
God bless ....I hope the explanation is useful.
