How to plot the implicit function $y^2=x^3/(a-x)$?

Question

How can I plot this implicit function: $$y^2=\dfrac{x^3}{a-x}$$

Should I use a polar coordinate transformation?

Answer 1

This is another command via Maple for doing that plot for any certain value of $a$:

$[>$ with(plots):

$[>$ implicitplot({seq($x^3/(a-x)-y^2 = 0$, $a = 1 .. 14$)}, $x = 0 .. 10, y = -20 .. 20$, numpoints $= 20000$);

Above I chose $a=14$, you can choose it $40$. :)

Answer 2

You can just rewrite it as $$y = \pm \sqrt{\dfrac{x^3}{a-x}}$$ and plot the positive and negative branches of that (which will be just mirror images of each other). Here's a quick example I did in Maple, with $a = 1$:

Note that, as the plot shows, your equation only has real solutions in the range $0 \le x < a$ (or $a < x \le 0$ if $a < 0$). Outside that range, $\frac{x^3}{a-x}$ will be negative (or undefined, for $x = a$), so $y$ would need to be imaginary.

In principle, you could also solve the original equation for $x$ as a function of $y$ and plot that. However, that entails solving the cubic polynomial $x^3+y^2x-y^2a=0$, and while there are straightforward methods for doing that, the result will not be particularly pretty.