It is of the second degree, and is the general equation of that degree. If, in fact, we multiply it by an unknown λ, we can, by solving six simultaneous equations in the six unknowns λ, h, k, e, p, α, so choose values for these as to make the coefficients in the equation equal to those in any equation of the second degree which may be given. There is no failure of this statement in the special case when the given equation represents two straight lines, as in § 10, but there is speciality: if the two lines intersect, the intersection and either bisector of the angle between them are a focus and directrix; if they are united in one line, any point on the line and a perpendicular to it through the point are: if they are parallel, the case is a limiting one in which e and h² + k² have become infinite while e−2(h² + k²) remains finite. In the case (§ 9) of an equation such as represents a circle there is another instance of proceeding to a limit: e has to become 0, while ep remains finite: moreover α is indeterminate. The centre of a circle is its focus, and its directrix has gone to infinity, having no special direction. This last fact illustrates the necessity, which is also forced on plane geometry by three-dimensional considerations, of treating all points at infinity in a plane as lying on a single straight line.
Sometimes, in reducing an equation to the above focus and directrix form, we find for h, k, e, p, tan α, or some of them, only imaginary values, as quadratic equations have to be solved; and we have in fact to contemplate the existence of entirely imaginary conics. For instance, no real values of x and y satisfy x² + 2y² + 3 = 0. Even when the locus represented is real, we obtain, as a rule, four sets of values of h, k, e, p, of which two sets are imaginary; a real conic has, besides two real foci and corresponding directrices, two others that are imaginary.
In oblique as well as rectangular coordinates equations of the second degree represent conics.
19. The three Species of Conics.—A real conic, which does not degenerate into straight lines, is called an ellipse, parabola or hyperbola according as e <, = , or > 1. To trace the three forms it is best so to choose the axes of reference as to simplify their equations.
In the case of a parabola, let 2c be the distance between the given focus and directrix, and take axes referred to which these are the point (c, 0) and the line x = − c. The equation becomes (x − c)² + y² = (x + c)², i.e. y² = 4cx.
In the other cases, take a such that a(e ~ e−1) is the distance of focus from directrix, and so choose axes that these are (ae, 0) and x = ae-1, thus getting the equation(x − ae)² + y² = e²(x − ae-1)², i.e. (1 − e²)x² + y² = a²(1 − e²). When e < 1, i.e. in the case of an ellipse, this may be written x²/a² + y²/b² = 1, where b² = a²(1 − e²); and when e > 1, i.e. in the case of an hyperbola, x²/a² − y²/b² = 1, where b² = a²(e² − 1). The axes thus chosen for the ellipse and hyperbola are called the principal axes.
In figs. 54, 55, 56 in order, conics of the three species, thus referred, are depicted.
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| Fig. 54 | Fig. 55 |
| Fig. 56. |
The oblique straight lines in fig. 56 are the asymptotes x/a = ±y/b of the hyperbola, lines to which the curve tends with unlimited closeness as it goes to infinity. The hyperbola would have an equation of the form xy = c if referred to its asymptotes as axes, the coordinates being then oblique, unless a = b, in which case the hyperbola is called rectangular. An ellipse has two imaginary asymptotes. In particular a circle x² + y² = a², a particular ellipse, has for asymptotes the imaginary lines x = ±y √−1. These run from the centre to the so-called circular points at infinity.