To determine your motion when you throw a basketball while you are standing on ice-skates; we use the principle of conservation of total momentum.

Let us assume the following:

your mass: M

the mass of the ball: Mb

the velocity of the ball when it is thrown: Vb in the horizontal direction

your velocity: V

We have assumed that the ball is thrown in a horizontal direction. The total momentum of the system which includes you and the ball was 0 initially, after you threw the ball it became Vb*Mb - V*b

So we arrive at Vb*Mb - V*b = 0

=> V = Vb*Mb / M

This means you are pushed back with a velocity equal to the product of the mass of the ball and the velocity of the ball divided by your mass.

The ball is pulled downward by the gravitation force of attraction equal to Mb*g after it is released. If it was held at a height of d when it was thrown, the time taken by the ball to strike the ground is sqrt (2*d/g). During this duration the ball has moved in the horizontal direction by a distance Vb*sqrt (2*d/g).

You follow a horizontal path after the ball is released, whereas the ball initially follows a parabolic path downwards. If we take the force of friction to be negligible, you continue to move in a straight line, and the ball after it strikes the ground starts to move in a straight horizontal line too.