HyperbolaGivenAsymptotics

Hyperbolas with given asymptotics (AC, AH) and passing through given point D.

To construct a hyperbola (c) with given asymptotic lines {AC, AH} and passing through a given point D.
Hyperbolas with given asymptotics and lying on the same angle-sectors defined by the given lines have the properties.
1] They are all homothetic with respect to their center at A.
2] From the asymptotics we know the ratio of their axes k = a/b = AB/BC.
3] From the point we get an additional equation involving a,b : (x/a)² - (y/b)² = 1.
4] From the two we determine a,b and c = sqrt(a²+b²).
5] From c we construct the two focal points.
6] Finally the conic with foci at G, G' and passing through the given D.
Since all these are homothetic they correspond to images of the same conic viewed from various distances.
This gives a good alternative to inspect a hyperbola in large distances or even in small instead of using magnifying glass.

Another way to construct the required hyperbola is by considering the family of hyperbolas on the same sector and with given asymptotics. A degenerate member of this family is the product of the two lines representing the asymptotes. Another member (c') can be constructed as shown.
- From an arbitrary point U on AB pass a circle centered at A.
- Project an intersection point R of this circle with AC to a point W on AB.
- Draw the hyperbola c' passing through W and having foci at U and its symmetric V w.r. to A

Hyperbolas with given asymptotics (AC, AH) and passing through given point D.

See Also