Construction of conic curves

The ellipse is the locus of the points such that the sum of their distances from two fixed points, called focal points or foci, is constant.

Let F and G be the focal points of an ellipse and s the sum of the distances of the points of an ellipse to the focal points. The following Descartes applet shows a point P which is restricted to move in such a way that

PF + PG = s

Exercises:

1) Drag the point P and you will see that the trail that it leaves is an ellipse. 
2) Change the value of s between 6 and 12 and draw the corresponding ellipses.
    What happens when s=6?

The hyperbola is the locus of points such that the difference of their distances to two fixed points, the foci, is constant. 

Let F and G be the focal points of a hyperbola and d  the difference of the distances of the points of the ellipse to the focal points.  The following applet shows a point P which is restricted to move in such a way that

PF - PG = d

Exercises:

1) Drag the point P and you will see that the trail it leaves is a hyperbola. 
2) Change the value of d between -5 and 5 and draw the corresponding hyperbolas.
    What happens when d=0?
    What happens when you change d for -d?

The parabola is the locus of the points such that their distances to a fixed point, called the focus and a fixed straight line, called the directrix, are equal.

Let F be the focus of a parabola and r its directrix.  The following applet shows a point P that is restricted to move in such a way that

PF = Pr

Exercises:

1) Drag the point P and you will see that the trail it leaves is a parabola. 
2) Change the distance Fr between 0 and 5 and draw the corresponding parabolas.
    What happens when Fr=0?

Author: José Luis Abru León

	Ministerio de Educación, Cultura y Deporte. Año 2000