You may also have a look at Nguyen Tan Tai's dakhiometric way to construct an hyperbola with compass and unmarked ruler:
1. Draw a circle.
2. Fix a point outside the circle.
3. Draw a segment between the fixed point and a point on that side of the circle which is the base of an isosceles triangle.
4. Draw height of isosceles and extend radius of circle, giving one side of isosceles. Height and extended radius intersect at vertex.
As the point runs on the half circle, the top vertex traces a hyperbola.