Equations of the conic curves.

Equations of conic curves with the vertex or centre at the origin.

The parabola.

The following applet presents the parabola with vertex at the origin and centre at the point(p,0). The equation appears above on the left in blue. The pupil can vary the value of p and see the appearance of the parabola with each value.

The following applet presents the parabola with vertex at the origin and centre at the point (0,p). The equation is shown above on the left, in blue.  The puil can vary the value of p and see the appearance of the parabola with each value.

The ellipse.

The following applet presents the ellipse with its centre at the origin and axes of symmetry coinciding with the coordinate axes. The horizontal axis is a and the vertical axis is b. The ellipse has its foci at the points (-c,0) and (c,0) where, c=sqrt(a^2-b^2), when b<a and at the points (0,c) and (0,-c), where c=sqrt(b^2-a^2), when a<b. The pupil can vary the values of a and b and see the appearance of the ellipse with each combination of values. When a=0 or b=0 the equation results in a division by zero which doesn´t have a meaning, so for this reason the values of a and b have been limited to be at least 0.01.

The hyperbola.

The following applet presents the hyperbola with its centre at the origin, axes of symmetry coinciding with the coordinate axes and with its focal points on the horizontal axis.  The horizontal axis is a and the vertical axis is b. The hyperbola has its focal points at the points (-c,0) and (c,0) where, c=sqrt(a^2+b^2). The asymptotes of the hyperbola are the straight lines which pass through the origin and have slopes b/a and -b/a. The pupil can vary the values of  a and b and see the appearance of the hyperbola with each combination of values. When a=0 or b=0 the equation presents a division by zero which doesn´t have a meaning, for this reason the values of a and b have been limited to be at least 0.01.

The following applet represents the hyperbola with centre at the origin, axes of symmetry coinciding with the coordinate axes and with its foci on the vertical axis.  As in the example above, the horizontal axis is a and the vertical axis is b, but now the focal points are on the vertical axis, on the points (0,c) and (0,-c). (c=sqrt(a^2+b^2) is the same as in the previous case). The asymptotes of the hyperbola are the same as those of the last example.

Equations of conic curves with variable vertex or centre.

The parabola.

The following applet presents the parabola with vertex in the point(h,k) and focus at (h+p,k). The equation appears at the top left in blue. The pupil can change the values of  p, h and k and observe the appearance of the parabola and the location of its vertex with each collection of values.

The ellipse.

The following applet presents the ellipse with centre at the point (h,k)and axes of symmetry parallel to the coordinate axes. The horizontal axis is a and the vertical axis is b. The ellipse has its focal points at  (h-c,k) and (h+c,k) where, c=sqrt(a^2-b^2), when b<a and at (h,k+c) and (h,k-c), where c=sqrt(b^2-a^2), when a<b. The pupil can vary the values of a, b, h and k and observe the aspect of the ellipse and the location of its centre and their foci with each set of values. When a=0 or b=0 the equation has a division by zero and makes no sense, for this reason the values of a and b have been limited to be at least 0.01.

The hyperbola.

The following applet presents the hyperbola with centre at(h,k)and axes of symmetry parallel to the coordinate axes. The horizontal semiaxes is a and the vertical one is b. The hyperbola has foci at (h-c,k) and (h+c,k) where, c=sqrt(a^2+b^2). The asymptotes of the hyperbola are the straight lines passing through (h,k) with slopes b/a and -b/a. The pupil can vary the values of a, b, h and k and observe the appearance of the hyperbola and the location of its centre, their foci and their asymptotes with each set of values. When a=0 or b=0 the equation has a division by zero and make no sense, for this reason the values of a and b have been limited to be at least 0.01.

Author: José Luis Abru León

	Ministerio de Educación, Cultura y Deporte. Año 2000