Locus and a Hyperbola

Author:
chisanga
Topic:
Hyperbola
The graph below shows a point O that is free to move along the hyperbola . The red line shows the intersection with the y-axis of the circle, with centre O and tangent to the x-axis. Note what happens to the distance between the intersections as O is moved.
What if O was on the portion of the curve that is below the x-axis; would the same rule apply?