the help of the following rule. Let A denote the discriminant of the equation; that is to say, the function abc+2fgh+af2 — bg2 — ch3, whose vanishing is the condition that the equation may represent right lines. Then A is the derived function formed from A, regarding a as the variable; and B, C, 2F, 2G, 2H are the derived functions taken respectively with regard to b, c, f, g, h. The coordinates of the centre (given Art. 140) may be written Ex. 1. Form the equation of the conic making intercepts A, A', μ, μ' on the axes. Since if we make y = 0 or x = 0 in the equation, it must reduce to the equation is x2 − (\ + λ) x + \\' = 0, y2 - (μ + μ') y + μμ' = 0; μμ' + 2hry + λλ' ? - μμ' (λ + λ 2 - λλ' (μ + μ) y + λλ'μμ' = 0, and his undetermined, unless another condition be given. Thus two parabolas can be drawn through the four given points; for in this case h = ± √(XX'μμ'). Ex. 2. Given four points on a conic, the polar of any fixed point passes through a fixed point. We may choose the axes so that the given points may lie two on each axis, and the equation of the curve is that found in Ex. 1. But the equation of the polar of any point x'y' (Art. 145) involves the indeterminate h in the first degree, and, therefore, passes through a fixed point. Ex. 3. Find the locus of the centre of a conic passing through four fixed points. The centre of the conic in Ex. 1 is given by the equations 2μp'x+2hy - μμ' (λ + λ') = 0, 2XX'y + 2hx − X\' (u + μ') = 0 ; whence, eliminating the indeterminate h, the locus is 2μμ'x2 - 2XX'у2 — uμ' (λ + λ') x + XX' (u + μ') y = 0, a conic passing through the intersections of each of the three pairs of lines which can be drawn through the four points, and through the middle points of these lines. The locus will be a hyperbola when X, X' and μ, u' have either both like or both unlike signs; and an ellipse in the contrary case. Thus it will be an ellipse when the two points on one axis lie on the same side of the origin, and on the other axis on opposite sides; in other words, when the quadrilateral formed by the four given points has a re-entrant angle. This is also geometrically evident; for a quadrilateral with a re-entrant angle evidently cannot be inscribed in a figure of the shape of the ellipse or parabola. The circumscribing conic must, therefore, always be a hyperbola, so that some vertices may lie in opposite branches. And since the centre of a hyperbola is never at infinity, the locus of centres is in this case an ellipse. In the other case, two positions of the centre will be at infinity, corresponding to the two parabolas which can be described through the given points. CHAPTER XI. EQUATIONS OF THE SECOND DEGREE REFERRED TO THE 152. IN investigating the properties of the ellipse and hyperbola, we shall find our equations much simplified by choosing the centre for the origin of coordinates. If we transform the general equation of the second degree to the centre as origin, we saw (Art. 140) that the coefficients of x and y will 0 in the transformed equation, which will be of the form. ax2 + 2hxy + by2 + c′ = 0. It is sometimes useful to know the value of c' in terms of the coefficients of the first given equation. We saw (Art. 134) that c' = ax2 + 2hx'y' + by'2 + 2gx' + 2ƒy' + c, where x, y are the coordinates of the centre. The calculation of this may be facilitated by putting into the form c' = (ax' + hy' + g) x' + (hx' + by' +ƒ) y' + gx' + fy' + c. The first two sets of terms are rendered =0 by the coordinates of the centre, and the last (Art. 140) 153. If the numerator of this fraction were = formed equation would be reduced to the form ax2 + 2hxy + by2 = 0, = 0, the trans and would, therefore (Art. 73), represent two real or imaginary Observing that when ƒ and g vanish the discriminant reduces to c (ab - h2), we can see that what has been here proved shows that transformation to parallel axes does not alter the value of the discriminant, a particular case of a theorem to be proved afterwards (Art. 371). It is evident in like manner that the result of substituting x'y', the coordinates of the centre, in the equation of the polar of any point "y", viz. (ax′ + hy' + g) x" + (hx' + by' +ƒ) y'" + gx' +ƒy' + c, is the same as the result of substituting x'y' in the equation of the curve. For the first two sets of terms vanish in both cases. right lines, according as ab-h' is negative or positive. Hence, as we have already seen, p. 72, the condition that the general equation of the second degree should represent two right lines, is abc+2fgh- af — bg2 — ch2 = 0. For it must plainly be fulfilled, in order that when we transfer the origin to the point of intersection of the right lines, the absolute term may vanish. Ex. 1. Transform 3x2 + 4xy + y2 — 5x - 6y - 30 to the centre (7, 4). Ex. 2. Transform x2 + 2xy - y2 + 8x + 4y − 8 = 0 to the centre (— 3, − 1). - Ans. x2+2xy - y2 = 22. 154. We have seen (Art. 136) that when satisfies the condition a cos*0+2h cos sin ✪ + b sin30 = 0, the radius vector meets the curve at infinity, and also meets the curve in one other point, whose distance from the origin is But if the origin be the centre, we have g=0, f=0, and this distance will also become infinite. Hence two lines can be drawn. through the centre, which will meet the curve in two coincident points at infinity, and which therefore may be considered as tangents to the curve whose points of contact are at infinity. These lines are called the asymptotes of the curve; they are imaginary in the case of the ellipse, but real in that of the hyperbola. We shall show hereafter, that though the asymptotes do not meet the curve at any finite distance, yet the further they are produced the more nearly they approach the curve. Since the points of contact of the two real or imaginary tangents drawn through the centre are at an infinite distance, the line joining these points of contact is altogether at an infinite distance. Hence, from our definition of poles and polars (Art. 89), the centre may be considered as the pole of a line situated altogether at an infinite distance. This inference may be confirmed from the equation of the polar of the origin, gx+fy+c=0, which, if the centre be the origin, reduces to c=0, an equation which (Art. 67) represents a line at infinity. 155. We have seen that by taking the centre for origin, the coefficients g and f in the general equation can be made to vanish; but the equation can be further simplified by taking a pair of conjugate diameters for axes, since then (Art. 143) h will vanish, and the equation be reduced to the form ax*+by+c=0. It is evident, now, that any line parallel to either axis is bisected. by the other; for if we give to x any value, we obtain equal and opposite values for y. Now the angle between conjugate diameters is not in general right; but we shall show that there is always one pair of conjugate diameters which cut each other at right angles. These diameters are called the axes of the curves and the points where they meet it are called its vertices. We have seen (Art. 143) that the angles made with the axis by two conjugate diameters are connected by the relation b tane taneh (tane + tan e') + a=0. But if the diameters are at right angles, tanē" =— (Art. 25). Hence h tan❜0+ (a - b) tan 0 — h = 0. 1 tan @ We have thus a quadratic equation to determine . Multiplying by p', and writing x, y, for p cose, p sine, we get hx2 - (a - b) xy — hy2 = 0. This is the equation of two real lines at right angles to each other (Art. 74); we perceive, therefore, that central curves have two, and only two, conjugate diameters at right angles to each other. On referring to Art. 75 it will be found that the equation. which we have just obtained for the axes of the curve is the same a that of the lines bisecting the internal and external angles between the real or imaginary lines represented by the equation ax* +2hxy+by2 = 0. The axes of the curve, therefore, are the diameters which bisect the angles between the asymptotes; and (note, p. 71) they will be real whether the asymptotes be real or imaginary; that is to say, whether the curve be an ellipse or a hyperbola. 156. We might have obtained the results of the last Article by the method of transformation of coordinates, since we can thus prove directly that it is always possible to transform the equation to a pair of rectangular axes, such that the coefficient of xy in the transformed equation may vanish. Let the original; axes be rectangular; then, if we turn them round through any angle 0, we bave (Art. 9) to substitute for x, x cos y sin 0, and for y, x sin0+ y cose; the equation will therefore become a (x cose-y sine)+2h (x cose-y sine) (x sine + y cose) + b (x sino + y cos 0)2 + c = 0 or, arranging the terms, we shall have the new a = a cos*+2h cos sin 0 + b sin30; the new hb sine cose + h (cos" - sin') - a sine cose; the new b = a sin30 - 2h cose sine + b cos* 0. Now, if we put the new h=0, we get the very same equation as in Art. 155, to determine tane. This equation gives us a simple expression for the angle made with the given axes by either axis of the curve, namely, 157. When it is required to transform a given equation to the form ax2+by+c=0, and to calculate numerically the value of the new coefficients, our work will be much facilitated by the following theorem: If we transform an equation of the second degree from one set of rectangular axes to another, the quantities a+b and ab-h' will remain unaltered. The first part is proved immediately by adding the values of the new a and b (Art. 156), when we have a' + b' = a + b. To prove the second part, write the values in the last article 2a = a+b+2h sin 20+ (a - b) cos 20, Hence 26′ = a+b-2h sin 20 - (a - b) cos 20. 4a′b′ = (a + b)2 — {2h sin 20 + (a − 3) cos20}3. 4h={2h cos 20 (a - b) sin20}; therefore 4 (a'b' - h’2) = (a + b)2 — 4h2 — (a — b)2 = 4 (ab — h3). When, therefore, we want to form the equation transformed to the axes, we have the new h=0, a'+b' = a + b, a'b' = ab- h2. |