In some places i see a being reserved for x axis and b for y axis, but others uses a for y axis if the ellipse is vertical or the hyperbola opens up and down.

Question

a and b use in ellipse and hyperbola

Answer 1

It depends on the direction of the semi-major and semi-minor axes, a and b, respectively.  Depending on the particular conic section, a could be parallel to the x-axis, the y-axis, or some other direction.  There is no convention that a has be horizontal and b has to be vertical.