You can try curves of the form $y = \dfrac{x^{n}}{(\alpha-x)^m}, \text{ where $x \in [0,\alpha)$}$ The parameters $m$ and $n$ controls how steep you want your growth to be and the parameter $\alpha$ controls where you want your curve to go to infinity. Below are some such curves for different values of $m$ and $n$ with $\alpha$ fixed as $5$. 
Now given $B$ on the curve, compute the length of the curve from $A$ to $B$, the length of the curve is given by $L_B = \int_{x_A}^{x_B} \sqrt{1 + \left(\dfrac{dy}{dx} \right)^2} dx$ Similarly, compute the length of the curve till $C$ which is given as $L_C = \int_{x_A}^{x_C} \sqrt{1 + \left(\dfrac{dy}{dx} \right)^2} dx$
Say you want $C$ to be at a distance such that $\dfrac{L_C}{L_B} = \dfrac{\beta}{100}$ where $\beta$ is the $\%$ distance of $C$ on the curve from $A$ to $B$. Once you compute $L_c$ from above, find $x_c$ such that $L_C = \displaystyle \int_{x_A}^{x_C} \sqrt{1 + \left(\dfrac{dy}{dx} \right)^2} dx$