Intermediate Level

Science and Technology

After the lines and circumferences, the ellipses, parabolas and hyperboles are the simplest lines. The fact that they can be obtained as sections of a conical surface of revolution is at the origin of the common designation of conics.

Their geometric characteristics give them a prominent role in the field of science and technology. Their association with the formulas that describe them establishes an alliance of Geometry with Algebra: the conics can be expressed by a second degree equation in the (x, y) coordinates.

$ax^2 + bxy + cy^2 + dx + ey + f = 0$

This application is intended to observe the effect of the variation of the parameters in this equation to obtain the different types of conics, depending on whether the value is positive, negative or null.

We suggest the exploration of situations in which the axes of the conics are parallel to the coordinated axes and the understanding of the meaning of the parameters in these cases.

It is due to Apollonius (262 BC, 190 BC) the definition of all conics from a single conical surface, as we now define them, and the assignment of the names that we continue to use: ellipse, parabola, and hyperbole.

 If the secant plane cuts out all the generatrixes of the cone, the resulting curve is an ellipse. If the secant plane is parallel to a single cone generatrix, the resulting curve is a parabola. If the secant plane is parallel to two cone generatrixes the resulting curve is a hyperbole.

Pappus, about five centuries later, studied the properties that allow to define the conics from the foci and the directrixes.

 Ellipse is the locus of the points of the plane such that the sum of the distances at two fixed points, the foci of the ellipse, is a constant greater than the distance between the foci. Parabola Is the locus of the points of the plane that are equidistant from a fixed point, the focus, and from a fixed line, the diretrix. Hyperbole is the locus of the points of the plane such that the module of the difference of the distances at two fixed points of the plane, the foci of the hyperbola, is a constant smaller than the distance between the foci.

In the seventeenth century it was shown, by work of Descartes and Fermat, that:

The whole second degree equation in the variables x and y, $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$, represents a conic (eventually degenerate at a point or two lines) and, conversely, the whole conic is represented by a second degree equation in x and y.

The type of conic depends on the value of $$b^2 - 4ac$$:

• Ellipse if$$b^2 - 4ac < 0$$;
• Parabola if $$b^2 - 4ac = 0$$;
• Hyperbole if $$b^2 - 4ac > 0$$.

It was Fermat who discovered the simplest equations of the ellipse (in particular the circumference), the parabola and the hyperbole:

 Ellipse: $${x^2\over{a^2}} + {y^2\over{b^2}} = 1$$ Parabola: $$x^2 = ay$$ Hyperbole: $${x^2\over{a^2}} - {y^2\over{b^2}} = 1$$