Algebraic Equation
This activity belongs to the GeoGebra book GeoGebra Principia.
Previously, we directly defined the distance from a point X(x,y) to the circle c as:
Xc(x,y):= Distance(X, c)
With this, the equation for the points equidistant from a point A and a circle reduces to:
XA – Xc = 0
If the circle has a center O and radius s, we can redefine the equation as:
|XO – s| = XA
This redefinition allows us to visualize the two branches of the hyperbola. To achieve this, we transform the previous irrational equation into an algebraic equation by squaring it to eliminate the roots (reaching the following expression is straightforward, as it doesn't require grouping, simplification or cancellation steps, but it's also okay to assist students with limited algebraic resources in solving this small exercise—the result is worthwhile):
(XA² – XO² – s²)² = 4s² XO²
Moreover, algebraic equations have the advantage of enabling representation of the corresponding inequalities without resorting to the offset method. For this, we simply define:
XA2(x,y) := Simplify(XA^2)
XO2(x,y) := Simplify(XO^2)
This way, we can introduce the inequalities:
(XA2 – XO2 – s²)² < 4s² XO2
(XA2 – XO2 – s²)² > 4s² XO2
Author of the construction of GeoGebra: Rafael Losada.