Usage of parameter "b" of an elliptic curve over GF(p)
Adding two points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \neq x_2$ gives point $(x_3, y_3)$ with

$$x_3 = (\frac{y_2 - y_1}{x_2 -x_1})^2- x_1 - x_2$$

$$y_3 = (\frac{y_2 - y_1}{x_2 -x_1}) (x_1 - x_3) - y_1$$

The case $x_1 = x_2$ is either adding the inverse or point doubling.

Doubling point $(x_1, y_1)$ with $y_1 \neq 0$ gives point $(x_2, y_2)$ with

$$x_2 = (\frac{3 x_1 + a}{2 y_1})^2 - 2 x_1$$

$$y_2 = -(\frac{3 x_1 + a}{2 y_1}) (x_1 - x_2) - y_1$$

So addition does require neither $a$ nor $b$, whereas point doubling and exponentiation with a scalar number requires only $a$. Checking if both points belong to the same curve also only requires $a$.