#-3=(x-2y)(x-y)# represents a hyperbola having asymptotes
x = 2y and x = y#
Use conversion formula # (x, y)= r (cos theta, sin theta )#.
The polar form is
#3= ((-cos^2theta+3sin theta cos theta -4 sin^2theta)/r^2#
#=-(cos theta - 2 sin theta)(cos theta - sin theta))/r^2#
Explicitly,
#r = 1/3sqrt((cos theta-sin theta)(2 sin theta - cos theta))#
The asymptotes are now obtained using r = 0 at the ( meet of the
asymptotes ) center..
So, they are ( for pairs of opposite directions ) #theta = pi/4, 5/4pi #
and #theta = pi+tan^(-1)(1/2), pi + tan^(-1)(1/2)#
Note that #theta# for the hyperbola #in (tan^(-1)(1/2), pi/4)# and
#(pi+tan^(-1)(1/2), 5/4pi)#, for the respective branches, in #Q_1 and Q_3#.
graph{(x-y)(x-2y)+3=0 [-40, 40, -20, 20]}
