To find the radius of curvature at any point $(x,y)$, we usually find the value of $\dfrac{dy}{dx}=y_1$ ; then find value of $\dfrac{d^2y}{dx^2}=y_2$ . Then we put the values to the equation
$$ρ=\dfrac{1}{\kappa}=\frac{(1+ y_1^2)^\frac{3}{2}}{y_2}$$
But for equations like $3x^2+4y^2=2x$ we have to follow a different approach like this [https://math.stackexchange.com/a/2588069/940998].
But then, for equation like $x^3+y^3=3axy$, the methods stated above doesn't work and needs another approach. So, how do we know beforehand which method to select?
HINT:
For the second curve (Folium of Descartes, please sketch or google its image) we have a double point at the origin.
Implicit differentiation leads to ($\text{curvature}= \kappa = \dfrac{d \phi}{d s})$
$$ (x^2 \cos \phi +y^2 \sin\phi )= a (x \sin \phi+ y \cos \phi) $$
$$( 2x \cos \phi -x^2 \sin \phi \;\kappa + 2 y \sin^2 \phi +y^2 \cos \phi\; \kappa)= a (x \sin \phi+ y \cos \phi) $$ Please check and write $\phi$ as subject of equation on LHS.
Now find $\kappa$ for both values $\phi= 0, \pi/2.$