How to trace the curve #(x^2+y^2)x=ay^2, a>0# stating all the properties used in the process?

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Answer 1 · 2017-05-22T10:08:32

See below

The best way is to do it with the help of polar coordinates.

#{(x = r costheta),(y=r sintheta):}#

substituting

#r^2cdotrcostheta=a r^2sin^2theta# or for #r gt 0#

#r=a sintheta tantheta#

we can compute easily some values for #r# given #theta#

#((theta,0,pi/4,pi/2),(r,0,sqrt(2)/2,oo))#

We know also that the curve is symmetric regarding the #x=0# axis having also a vertical asymptote at #x = a# because

#y^2=x^3/(a-x)#

Attached a plot for #a = 2#

enter image source here