1 III. Let the line OO' be a diameter, and OR, O'p, parallel to its ordinates, then OR' = OR" and O'p' = O'p". Let the diame ter meet the curve in the points A, B, then OR2 = AO.OBAO.O'B' Hence, The squares of the ordinates of any diameter are proportional to the rectangles under the segments which they make on the diameter. 153. There is one case in which the theorem of Article 151 becomes no longer applicable, namely, when the line OS is parallel to one of the lines which meet the curve at infinity; the segment OS" is then infinite, and OS only meets the curve in one finite point. We propose, in the present Article, to inquire OS' whether, in this case, the ratio OR'. OR" will be constant. Let us, for simplicity, take the line OS for our axis of x, and OR for the axis of y. Since the axis of x is parallel to one of the lines which meet the curve at infinity, the term A will = 0 (Art. 136(3)), and the equation of the curve will be of the form Bxy + Cy2+ Dx + Ey + F OS' = 0. Making y = 0, the intercept on the axis of x is found to be ; and making ≈ = 0, the rectangle under the intercepts Now, if we transform the axes to any parallel axes (Art. 129), C will remain unaltered, and the new D = By' + D. Hence the new ratio will be C Now, if the curve be a parabola, B = 0, and this ratio is constant; hence, if a line parallel to a given one meet any diameter (Art. 140) of a parabola, the rectangle under its segments is in a constant ratio to the intercept on the diameter. If the curve be a hyperbola, the ratio will only be constant while y' is constant; hence the intercepts made by two parallel chords of a hyperbola, on a parallel to an asymptote, are proportional to the rectangles under the segments of the chords. *154. To find the condition that the line ax + by + c = 0 should touch the conic represented by the general equation. Solving for y from ax + by + c = 0, and substituting in the general equation, the abscissæ of the points where this line meets the conic are determined by the quadratic (Ab2-Bab+ Ca2) x2 - (Bbc-2Cac-Db2+ Eab)x+Cc2 - Ebc + Fb2 = 0. If the line touch the conic, this quadratic will have equal roots, or (Bbc - 2Cac - Db2 + Eab)2 = 4 (Ab2 – Bab + Ca2) (Cc2 – Ebc + Fb2). Multiplying out, this equation becomes divisible by b2, and may be arranged (E2 – 4CF) a2 + (D2 – 4AF) b2 + (B2 – 4AC) c2 + 2 (2AE – BD) bc + 2 (2CD – BE) ca + 2 (2BF – DE) ab MISCELLANEOUS EXAMPLES. = 0. Ex. 1. To find the equation of the conic which makes intercepts a, a', b, b', on the axes. The intercepts on the axes are given by the quadratics x2 − (a + a') x + aa′ = 0, y2 − (b + b′) y + bb' = 0, but these must be what the general equation becomes when in it we make y = 0, x = 0 ; hence the equation is bb'x2 + Bxy + aa'y2 – bb′ (a + a′) x − aa′ (b + b′) y + aa′bb' = 0, where B is still indeterminate. Ex. 2. To find the equation of the parabola which touches the axes at points x = a, y = b. In the preceding make a = a', b = b', and determine B by the condition B2 = 4AC, and we find b2x2 - 2abxy + a2 y2 — 2b2 ax − 2a2by + a2b2 : = 0. to the coefficient of xy, since if we gave the sign + it would not re present a parabola, but the square of the line bx+ay - ab = 0. Ex. 3. Given four points on a conic, the polar of any fixed point passes through a fixed point. Take for axes two opposite sides of the quadrilateral formed by the points; then form by Art. 144 the polar of x'y' with regard to the conic found in Ex. 1, and it will contain the indeterminate B in the first degree, and therefore passes through a fixed point. Ex. 4. Find the locus of the centre of a conic passing through four given points. The centre of the conic in Ex. 1 is given by the equations 2bb'x + By - bb' (a + a') = 0, · ́2aa'y + Bx − aa' (b + b') = 0. T Eliminate the indeterminate B, and the locus is 2bb'x2 - 2aa'y2 bb' (a + a') x − aa' (b + b') y = 0, a conic passing through the intersections of each of the three pair of lines which can be drawn through the four points, and through the middle points of those lines. CHAPTER XI. EQUATIONS OF THE SECOND DEGREE REFERRED TO THE CENTRE AS ORIGIN. 155. In investigating the properties of the ellipse and hyperbola, we shall find our equations much simplified by choosing the centre for the origin of co-ordinates. If we transform the general equation of the second degree to the centre as origin, we saw (Art. 138) that the coefficients of x and y will = 0 in the transformed equation, which will be of the form Ax2 + Вxy + Cy2 + F′ = 0. It is sometimes useful to know the value of F' in terms of the coefficients of the first given equation. We saw (Art. 129) that F' = Ax22 + Bx'y' + Cy'2 + Dx' + Ey' + F, where x', y', are the co-ordinates of the centre. The calculation of this may be facilitated by putting F' into the form F' = {(2Ax' + By' + D) x' + (2Cy' + Bx' + E) y' + Dx' + Ey' + 2F}. The first two terms must be rendered of the centre, and the last (Art. 138) = O by the co-ordinates 156. If the numerator of this fraction were = 0, the transformed equation would be reduced to the form Ax2+ Bxy + Cy2 = 0, and would, therefore (Art. 69), represent two real or imaginary right lines, according as B2 - 4AC is positive or negative. Hence, as we have already seen, p. 67, the condition that the general equation of the second degree should represent two right lines, is AE2+ CD2 + FB2 - BDE-4ACF = 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 Ans. 12x2 + 16xy+4y2 + 1 = 0. Ex. 2. Transform x2 + 2xy - y2 + 8x + 4y − 8 = 0 to the centre (— 3, − 1). Ans. x2+2xy — y2 = 22. 157. We have seen (Art. 134) that the equation Ax2+ Bxy + Cy2 = 0 represents the real or imaginary lines drawn through the origin to meet the curve at infinity; and that each of these lines will meet the curve in one other point, at a distance from the origin, But if the origin be the centre, we have D = 0, E = 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 that 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. 143) 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, Dx + Ey + 2F = 0, which, if the centre be the origin, reduces to F = 0, an equation which (Art. 64) represents a line at infinity. 158. We have seen that by taking the centre for origin the coefficients D and E 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. 141) B will vanish, and the equation be reduced to the form 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 curve, and the points where they meet it are called its vertices. The equation of the diameter conjugate to my = nx is (Art. 141); and this will be perpendicular to my = nx (Art. 40) (2Am + Bn) n − (Bm + 2Cn) m = 0, if or Bm2 - 2(A - C) mn - Bn2 = 0; or, multiplying by p2, and writing x, y for mp, np, Bx2 - 2(AC) xy – By2 = 0. This is the equation of two real lines at right angles to each other (Art. 70); we perceive, therefore, that central curves have two, and only two, conjugate diameters at right angles to each other. On referring to Art. 71 it will be found, that the equation. which we have just obtained for the axes of the curve is the same as that of the lines bisecting the internal and external angles between the real or imaginary lines represented by the equation Ax2 + Bxy + Cy2 = 0. The axes of the curve, therefore, are the diameters which bisect the angles between the asymptotes; and (note, p. 66) they will be real whether the asymptotes be real or imaginary: that is to say, whether the curve be an ellipse or an hyperbola. 159. We might have obtained the results of the last Article by the method of transformation of co-ordinates, since we can thus prove directly that it is always possible to transform the |