137. The most important question we can ask, concerning the form of the curve represented by any equation, is, whether it be limited in every direction, or whether it extend in any direction to infinity. We have seen, in the case of the circle, that an equation of the second degree may represent a limited curve, while the case where it represents right lines shows us that it may also represent loci extending to infinity. It is necessary, therefore, to find a test whereby we may distinguish which class of locus is represented by any particular equation of the second degree.

With such a test we are furnished by the last article. For if the curve be limited in every direction, no radius vector drawn from the origin to the curve can have an infinite value; but we found in the last article, that when the radius vector becomes infinite, we have a + 2h tan0+b tan2 0 = 0.

(1) If now we suppose h2 - ab to be negative, the roots of this equation will be imaginary, and

no real value of 0 can be found which will render

a cos2+2h cos✪ sin ✪ +b sin2 0 = 0. In this case, therefore, no real line can be drawn to meet the curve at infinity, and the curve will be limited

X

in every direction. We shall show, in the next chapter, that its form is that represented in the figure. A curve of this class is called an Ellipse.

(2) If h2 – ab be positive, the roots of the equation

a+2h tan + b tan2 0 = 0

will be real; consequently, there are two real values of which will render infinite the radius vector to the curve. Hence, two real lines (ax2 + 2hxy + by2 = 0) can, in this case, be drawn through the origin to meet the curve at infinity. A curve of this class is called an

X

Hyperbola, and we shall show, in the next chapter, that its form is that represented in the figure.

(3) If h2 — ab = 0, the roots of the equation

a+2h tan + b tan2 0 = 0

will then be equal, and, therefore, the two directions in which a right line can be drawn to meet the curve at infinity will in this case coincide. A curve of this class is called a Parabola, and we shall (Chap. XII.) show that its form is

Y

X

that here represented. The condition here found may be otherwise expressed, by saying that the curve is a parabola when the first three terms of the equation form a perfect square.

138. We find it convenient to postpone the deducing the figure of the curve from the equation, until we have first by transformation of co-ordinates, reduced the equation to its simplest form. The general truth however of the statements. in the preceding article may be seen if we attempt to construct the figure represented by the equation, in the manner explained (Art. 16). Solving for y in terms of x, we find (Art. 76)

by = − (hx +ƒ) ± √√ {(h2 — ab) x2 + 2 (hf − bg) x + (ƒ2 — bc)}. Now, since by the theory of quadratic equations, any quantity of the form x2+px + q is equivalent to the product of two real or imaginary factors (x − a) (x-B), the quantity under the radical may be written (h2 — ab) (x − a) (x − ß). If then hab be negative, the quantity under the radical is negative, (and therefore y imaginary), when the factors x-α, x— ß are either both positive, or both negative. Real values for y are only found when x is intermediate between a and B, and therefore. the curve only exists in the space included between the lines x=α, x=ẞ (see Ex. 3, p. 13). The case is the reverse when h2- ab is positive. Then we get real values of y for any values of x, which make the factors x-a, x-B either both positive or both negative; but not so if one is positive and the other negative. The curve then consists of two branches stretching to infinity both in the positive and in the negative direction, but separated by an interval included by the lines =α, x=ß, which no part of the curve is found. If h-ab vanishes, the

quantity under the radical is of the form either x

a or a

In the one case we have real values of y, provided only that x is greater than a; in the other, provided only that it is less. The curve therefore consists of a single branch stretching to infinity either on the right or the left-hand side of the line x = a. If the factors a and ẞ be imaginary, the quantity under the radical may be thrown into the form (ab) {(x − y)2 + 82}. ← If then hab is positive, the quantity under the radical is always positive, and lines parallel to the axis of y always meet the curve. Thus in the figure of the hyperbola, p. 135, lines parallel to the axis of y always meet the curve, although lines parallel to the axis of x may not. On the other hand, if h2 — ab is negative, the quantity under the radical is always negative, and no real figure is represented by the equation.

Ex. 1. Construct, as in Art. 16, the figures of the following curves, and determine their species:

3x2 + 4xy + y2 — 3x - 2y + 21 = 0.
5x2 + 4xy + y2 – 5x - 2y — 19 = 0.
4x2 + 4xy + y2 — 5x - 2y — 10 = 0.

Ex. 2. The circle is a particular case of the ellipse.

Ans. Hyperbola.

Ans. Ellipse.

Ans. Parabola.

For in the most general form of the equation of the circle, a = b, h = a cos∞ (Art. 81); and therefore h2 - ab is negative, being =- a2 sin2 w.

Ex. 3. What is the species of the curve when h=0? Ans. An ellipse when a and b have the same sign, and an hyperbola when they have opposite signs.

0;

Ex. 4. If either a or b0, what is the species? Ans. A parabola if also h otherwise a hyperbola. When a = 0 the axis of x meets the curve at infinity; and when b = 0, the axis of y.

Ans. A parabola touching the axes at the points x = a, y = b.

139. If in a quadratic Ax2+2Bx+C=0, the coefficient B vanishes, the roots are equal with opposite signs. This then will be the case with the equation

(a cos30+ 2h cos✪ şin 0 + b sin36) p2 + 2 (g cos✪ +ƒ sin☺) p+c=0, if the radius vector be drawn in the direction determined by the equation g cose +ƒ sin=0.

The points answering to the equal and opposite values of p are equidistant from the origin, and on opposite sides of it;

T

therefore, the chord represented by the equation gx+fy=0 is bisected at the origin.

Hence, through any given point can in general be drawn one chord, which will be bisected at that point.

140. There is one case, however, where more chords than one can be drawn, so as to be bisected, through a given point.

If, in the general equation, we had g=0, f=0, then the quantity g cose+f sine would be = 0, whatever were the value of 0; and we see, as in the last article, that in this case every chord drawn through the origin would be bisected. The origin would then be called the centre of the curve. Now, we can in general, by transforming the equation to a new origin, cause the coefficients g and f to vanish. Thus equating to nothing the values given (Art. 134) for the new g and ƒ, we find that the co-ordinates of the new origin must fulfil the conditions

ax' + hy' +g=0, hx' + by' +ƒ=0.

These two equations are sufficient to determine x' and y', and being linear, can be satisfied by only one value of x and y; hence, conic sections have in general one and only one centre. Its co-ordinates are found, by solving the above equations, to be

bg-hf
y'
h2 - ab'

In the ellipse and hyperbola h2 — ab is always finite (Art. 137); but in the parabola h2 — ab = 0, and the co-ordinates of the centre become infinite. The ellipse and hyperbola are hence often classed together as central curves, while the parabola is called a non-central curve. Strictly speaking, however, every curve of the second degree has a centre, although in the case of the parabola this centre is situated at an infinite distance.

141. To find the locus of the middle points of chords, parallel to a given line, of a curve of the second degree.

if

We saw (Art. 139) that a chord through the origin is bisected cose +ƒ sin00. Now, transforming the origin to any point, it appears, in like manner, that a parallel chord will be

g

bisected at the new origin if the new g multiplied by cose + the new f multiplied by sin0=0, or (Art. 134)

cos (ax+hy+g) + sin✪ (hx' + by' +ƒ)

= 0.

This, therefore, is a relation which must be satisfied by the coordinates of the new origin, if it be the middle point of a chord making with the axis of x the angle 0. Hence the middle point of any parallel chord must lie on the right line

cos 0 (ax+hy+g) + sin ✪ (hx+by+f) = 0,

which is, therefore, the required locus.

Every right line bisecting a system of parallel chords is called a diameter, and the lines which it bisects are called its ordinates. The form of the equation shows (Art. 40) that every diameter must pass through the intersection of the two lines

Y

N

M

M'

X

ax+hy+g=0, and hx+by+f=0:
but, these being the equations by
which we determined the co-ordinates
of the centre (Art. 140), we infer that
every diameter passes through the centre of the curve.
It appears by making
alternately =0, and 90° in

=

is the equation of the diameter bisecting chords parallel to the

constructed by first laying down the line hx + by +f, and then taking on each ordinate MP of that line, portions PQ, PQ', above and below P and equal to R. Thus also it appears that each ordinate is bisected by hx + by + f.