[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/sw2cat9w]GeoGebra Principia[/url].[/color][br][br][br][color=#CC3300][b]Implicit Curves from Definitions in CAS[/b][/color][br][br]A parabola can be defined as the locus of points in the plane equidistant from a line (directrix) and an external point (focus). Locating one point (the vertex) is easy, but how do we locate the others?[br][br]With GeoGebra, we can create a free point to explore the situation and mark those positions where both distances are equal. It's [u]quite instructive[/u] but, after several exercises, it becomes tedious.[br][br]Alternatively, we can construct a generic point that defines the locus, but this construction will only work for this case or similar cases.[br][br]We can also create the [b]implicit curve[/b] by defining an arbitrary point [b]X(x,y)[/b] in the [b]CAS View[/b]:[br][br] [color=#CC3300]X:= (x, y)[/color][br][br]the distance from [b]X[/b] to the focus [b]F[/b]:[br][br] [color=#CC3300]XF(x,y):= Distance(X, F)[/color][br]  [br]the distance from [b]X[/b] to the directrix [b]r[/b]: [br]  [br] [color=#CC3300]Xr(x,y):= Distance(X, r)[/color][br]  [br] and by equating both distances: [br]  [br] [size=150][color=#CC3300][b]XF – Xr = 0[/b][/color][/size][br]  [br]GeoGebra uses numerical algorithms to create this implicit curve, so small errors or omissions may appear in some cases.[br][list][*][color=#999999]Note: [i]At least for now[/i], GeoGebra GeoGebra does not represent equations of this type in three variables. That is, it recognizes x² + y² + z² = 16 as a sphere, but it does not recognize the equivalent equation (sqrt(x² + y² + z²))² = 16 as such.[/color][/*][/list]

[color=#999999]Author of the construction of GeoGebra: [url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color]