Solution

$$xy^k =c^2$$
$$\Rightarrow \ x.xy^{k-1} \dfrac {dy}{dx}+y^k =0$$
$$\Rightarrow \ \left (\dfrac {x}{y}\right) k\ y^{k} \dfrac {dy}{dx}+y^k =0$$
$$\Rightarrow \ y^k \left (\dfrac {kx}{y} \dfrac {dy}{dx}+1\right)=0$$
$$\therefore \ \boxed {y=0}$$ or $$\dfrac {kx}{y}.\dfrac {dy}{dx}=-1$$
$$\left (\dfrac {dy}{dx}=\dfrac {-y}{kx}\right)$$
Length of subtangent $$=y\dfrac {dx}{dy}=y.\dfrac {kx}{-y}=-kx$$
Also, length of subtangent = constant (independent of $$r$$)
$$\therefore \ \boxed {k=0}\ (A)$$