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

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