Step 1: Special case check
Try $m = 2$: $x^2 + y^2 = k$ is a circle. The second derivative of a circle is $y'' = -\dfrac{k}{y^3}$.
Step 2: Match
Compare with $\dfrac{a x^b}{y^c}$: $a = -k$, $b = 0$, $c = 3$. Then $a + b + c = -k + 3 = 0$, so $k = 3$.
Step 3: General confirmation
Doing the same with a general $m$, the sum $a + b + c$ factorises as $(m-1)(3-k)$, so the answer 3 holds for every $m \neq 1$.
Step 4: Result
$k = 3$, which is option (C).
Final Answer:
k equals 3. This is option (C).
\[ \boxed{\text{(C) }3} \]