33 CHAPTER II. SPECIAL FORMS OF THE EQUATION OF THE SECOND DEGREE. 1. WE now proceed to the discussion of the curve represented by the equation of the second degree. We shall first prove that every curve, represented by such an equation, is what is commonly called a conic section; and then, before proceeding further with the consideration of the general equation, shall investigate the nature of the curve corresponding to certain special forms of the equation. PROP. Every curve represented by an equation of the second degree is cut by a straight line in two points, real, coincident, or imaginary. The general equation of the second degree is represented by ua2 + vß2 + wy2+2u' By + 2v' ya+2w' aß=0. To find where the curve, of which this is the equation, is cut by the straight line la+mẞ+ny = 0, we may eliminate a between the two equations. This will give us a quadratic for the determination of, to each of the two values of this ratio, real, equal, or imaginary, one value of a will correspond; whence it appears that the straight line and the curve cut one another in two real, coincident, or imaginary points. Hence, the curve is of the same nature as that represented by the equation of the second degree in Cartesian co-ordinates, and is, therefore, a conic section. 2. We shall now inquire what are the relations of the conic section to the triangle of reference, when certain relations exist among the coefficients of the equation. First, suppose u, v, w, all =0. The equation then assumes the form u'ßy + v'ya + w'aß = 0, which we shall write λβγ + μγα + ναβ = 0. Now, if in this equation we put a = 0, it reduces itself to λβγ = 0, which requires either that ß=0, or that y = 0. It hence appears that the curve passes through two of the angular points (B, C) of the triangle of reference. It may similarly be shewn to pass through the third. Hence the equation λβγ + μγα + ναβ = 0, or, as it may also be written, μ V represents a conic, described about the triangle of reference. 3. Let us now inquire how the line βγ μ ע If in the equation of the conic we put + = 0, or, which is the same thing, uy+vß= 0, it reduces to λBy=0. Hence the line t βγ ע =0 meets the conic in the points in which it meets the lines = 0, y=0; but these two points coincide, since the line in question evidently passes through the point of intersection of B=0 and y=0. Hence the straight line and the conic meet one another in coincident points, that is, they touch one another at the point A. CENTRE OF THE CONIC. Similarly, the equations of the tangents at B and C are 35 4. To determine the position of the centre of the conic. Through the angular points A, B, C of the triangle of reference draw the tangents EAF, FBD, DCE. Bisect Fig. 14. F E H B Q AC, AB respectively in H, I, join EH, FI, and produce them to intersect in O. Then, since every straight line drawn through the intersection of two tangents so as to bisect their chord of contact passes also through the centre, O will be the centre of the conic. These equations determine the position of the centre. COR. We may hence deduce the relation which must hold between λ, u, v, in order that the conic may be a parabola. For, since the centre of a parabola is at an infinite distance, its co-ordinates will satisfy the equation aa+bB+cy=0. We hence obtain the following equation: X2a2 + μ3b2 + v2c2 - 2 μvbc — 2vλca — 2λμ ab = 0, CONDITION FOR A PARABOLA. 37 which is equivalent to ± (λa)3 ± (μb)* ± (vc)* = 0, as the necessary and sufficient condition that the conic should be a parabola. 5. To determine the condition that a given straight line may touch the conic. If the conic be touched by the straight line (7, m, n), the two values of the ratio ẞy, obtained by eliminating a between the equations λβγ + μγα + ναβ = 0, la + mB + ny = 0, must be coincident. The equation which determines these is -XBY + (uy +vß) (mß +ny) = 0, and the condition that the two values of ß: y be equal, is 4μn.vm — (μm + vn — Xl)2 = 0, or λ3l2 + μ3m2 + v3n2 — 2μv. mn − 2vλ . nl — 2λμ. Im = 0, which is equivalent to ± (~7)3 ± (μm)* ± (vn)* = 0. If this be compared with the condition investigated in Art. (4) that the conic may be a parabola, it will be observed that the parabola satisfies the analytical condition of touching the straight line ax + bß + cy=0. This is generally expressed by saying that every parabola touches the line at infinity. 6. To investigate the equation of the circle, circumscribing the triangle of reference. This may be deduced from the consideration that the co-ordinates of the centre of the circumscribing circle are respectively proportional to cos A, cos B, cos C (see p. 4). Or |