I want to plot the following
$$L(\rho) = m + \frac{c}{\rho^2}$$
That's very straightforward.
y6[m_, c_, \[Rho]_] := m + c/\[Rho]^2
Plot[{y6[1, -2, r], y6[1, -0.5, r], y6[1, -0.1, r], y6[1, 0, r],
y6[1, 0.1, r], y6[1, 0.5, r], y6[1, 2, r]}, {r, 1, 10},
PlotLegends -> "Expressions", PlotRange -> {0.76, 1.25},
AxesOrigin -> {1.2, 0.84}]
The ($L$ vs $\rho$)-plot is
A valid change of coordinates is the following
$$\rho = r \sin(\theta)$$ $$L = r \cos(\theta)$$
And the corresponding plot is the $r \cos(\theta)$ vs $r \sin(\theta)$.
I don't know how to implement correctly the change of coordinates and how to write the command to get the $r \cos(\theta)$ vs $r \sin(\theta)$ plot.