Draw two points on the drawing pad using the ‘New Point’ tool. Use the ‘slider’ tool and define two sliders, say d and r.
Use the “Circle with centre and radius’ tool and draw a circle with A as the centre and r as the radius. Draw another circle with B as the centre and d +r as the radius. Use the intersect tool and find the intersection points of these two circles. Named as point C and point D.
Use the ‘Perpendicular bisector’ tool and find the perpendicular bisector of AB.
We know that the two arms of the hyperbola are mirror images of each other when reflect along the perpendicular bisector of the line segment between the two focii. So we will reflect the construction to get the second arm.
Use the ‘Reflect about line’ tool and reflect the point C and point D along the perpendicular bisector.
Now you have two points of intersection and their images. Right click on all the four and put their trace on.
Keep the slider d constant. Animate the slider r by right clicking on that and putting on the animation.

Try out different ways to make hyperbola using GeoGebra.