CA sponding quantities for the other pairs of tangents are A' B' = and these three multiplied together are 1. Hence, recollecting the meaning of k (Art. 54), we learn that if A, F, B, D, C, E be the angles of a circumscribing hexagon, = 1. sin EAB.sin FAB.sin FBC. sin DBC.sin DCA. sin ECA sin EAC.sin FA C. sin FBA.sin DBA.sin DCB.sin ECB Hence also three pairs of lines will touch the same conic if their equations can be thrown into the form M2+ N*+ 2ƒ'MN=0, N3+ L3+ 2g'NL=0, L2 + M2 + 2h'LM = 0, for the equations of the three pairs of tangents, already found ean be thrown into this form by writing L√(4) for a, &c. 296. If we wish to form the equations of the lines joining to a''y all the points of intersection of two curves, we have only to substitute la + ma', 13 + mẞ', ly + my in both equations, and eliminate 1: m from the resulting equations. For any point on any of the lines in question evidently possesses the property that the line joining it to a''y meets both curves in the same point; therefore the equations in 7: m, which determine the points where one of these lines meets both curves, must have a common root; and therefore the result of elimination between them is satisfied. Thus, the equation of the pair of lines joining to a'B'y' the points where any right line L meets S, is L'S-2LLP+ L'S=0. If the point a'B'y' be on the curve the equation reduces to L'S-2LP=0. Ex. A chord which subtends a right angle at a given point on the curve passes through a fixed point (Ex. 2, Art. 181). We use the general equation, and by the formula last given, form the equation of the lines joining the given point to the intersection of the conic with λx+uy+v. The coordinates being supposed rectangular, these lines will be at right angles if the sum of the coefficients of x2 and y2 vanish, which gives the condition viz. And since λ, u, v enter in the first degree, the chord passes through a fixed point, b -a x', b+ a a+ b by. If the point on the curve vary, this other point will describe a conic. If the angle subtended at the given point be not a right angle, or if the angle be a right angle, but the given point not on the curve, the condition found in like manner will contain A, μ, v in the second degree, and the chord will envelope a conic. 297. Since the equation of the polar of a point involves the coefficients of the equation in the first degree, if an indeterminate enter in the first degree into the equation of a conic it will enter in the first degree into the equation of the polar. Thus, if P and P be the polars of a point with regard to two conics S, S', then the polar of the same point with regard to S+kS will be P+kP'. For (a + ka') aa' + &c. = aaa' + &c. + k {a'aa' + &c.}. Hence, given four points on a conic, the polar of any given point passes through a fixed point (Ex. 2, Art. 151). If Q and Q' be the polars of another point with regard to S and S', then the polar of this second point with regard to S+kS is QkQ. Thus, then (see Art. 59), the polars of two points with regard to a system of conics through four points form two homographic pencils of lines. Given two homographic pencils of lines, the locus of the intersection of the corresponding lines of the pencils is a conic through the vertices of the pencils. For, if we eliminate k between P+kP', Q+kQ, we get PQ'P'Q. In the particular case under consideration, the intersection of P+kP, Q+kQ is the pole with respect to S+kS' of the line joining the two given points. And we see that, given four points on a conic, the locus of the pole of a given line is a conic (Ex. 1, Art. 278). If an indeterminate enter in the second degree into the equation of a conic, it must also enter in the second degree into the equation of the polar of a given point, which will then envelope a conic. Thus, if a conic have double contact with two fixed conics, the polar of a fixed point will envelope one of three fixed conics; for the equation of each system of conics in Art. 287 contains μ in the second degree. We shall in another chapter enter into fuller details respecting the general equation, and here add a few examples illustrative of the principles already explained. Ex. 1. A point moves along a fixed line; find the locus of the intersection of its polars with regard to two fixed conics. If the polars of any two points a'ß'y', a"ß"y" on the given line with respect to the two conics be P', P"; Q', Q"; then any other point on the line is λa' + μa", \ß' +μß", λy' +μy"; and its polars λP' +μP" \Q' + μQ", which intersect on the conic P'Q" = P"Q'. Ex. 2. The anharmonic ratio of four points on a right line is the same as that of their four polars. For the anharmonic ratio of the four points la' + ma", l'a' + m'a", l''a' +m"a", l'"'a' +m""a", is evidently the same as that of the four lines IP' + mP", l'P' + m'P", l''P' +m"P", l'"'P' +m""P". Ex. 3. To find the equation of the pair of tangents at the points where a conic S is met by the line y. The equation of the polar of any point on y is (Art. 291) a'S, +ẞ'S2 = 0. But the points where y meets the curve are found by making y = 0 in the general equation, whence aa”? + 2ha + b” = 0. Eliminating a', B' between these equations, we get for the equation of the pair of tangents aS2-2hS2S2+b812 = 0. Thus the equation of the asymptotes of a conic (given by the Cartesian equation) is b a 2 (15)2 – 2h (ds) (ds) + 6 (ds)2 = 0, for the asymptotes are the tangents at the points where the curve is met by the line at infinity z. Ex. 4. Given three points on a conic: if one asymptote pass through a fixed point, the other will envelope a conic touching the sides of the given triangle. If ty, to be the asymptotes, and aa + bẞ+cy the line at infinity, the equation of the conic is t1t2 = (aa + bẞ+cy)2. But since it passes through ßy, ya, aß, the equation must not contain the terms a2, 62, y2. If therefore t1 be λa + μß + vy, to must a2 b2 be a+ B+ y; and if t2 pass through a'ß'y', then (Ex. 1, Art. 285) t, touches μ λ a √(aa') + b √(BB') + c J(yy') = 0. The same argument proves that if a conic pass through three fixed points, and if one of its chords of intersection with a coníc given b by the general equation aa2 + &c. = 0 be λa + μẞ+vy, the other will be a+ B+ α с Ex. 5. Given a self conjugate triangle with regard to a conic: if one chord of intersection with a fixed conic (given by the general equation) pass through a fixed point, the other will envelope a conic [Mr. Burnside]. The terms aß, ßy, ya are now to disappear from the equation, whence if one chord be λa + μß +vy, the other is found to be λa (ug + vh - Xƒ) + μß (vh + \ƒ — uμg) + vy (\ƒ + μg − vh). Ex. 6. A and A′ (α1ẞ1Y1, α2ß272) are the points of contact of a common tangent to two conics U, V; P and P' are variable points, one on each conic; find the locus of C, the intersection of AP, A'P', if PP' pass through a fixed point O on the common tangent [Mr. Williamson]. Let P and Q denote the polars of aẞ1Y1, a2ß2Y2, with respect to U and V respectively; then (Art. 290) if aßy be the coordinates of C, those of the point P where AC meets the conic again, are Uα, - 2Pα, Uß1 – 2Pß, Uɣı — 2Py; and those of the point P' are, in like manner, Va2 - 2Qa, &c. If the line joining these points pass through 0, which we choose as the intersection of a, ß, we must have == Ua, - 2Pa Va2- 2Qa and when A, A', O are unrestricted in position, the locus is a curve of the fourth order. If, however, these points be in a right line, we may choose this for the line a, and making a, and a2 = 0, the preceding equation becomes divisible by a, and reduces to the curve of the third order PVB2 = QUB1. Further, if the given points are points of contact of a common tangent, P and Q represent the same line; and another factor divides out of the equation which reduces to one of the form U kV, representing a conic through the intersection of the given conics. Ex. 7. To inscribe in a conic, given by the general equation, a triangle whose sides pass through the three points ßy, ya, aß. We shall, as before, write S1, S2, S3 for the three quantities, aa + hẞ + gy, ha + bß +fy, ga+fß+cy. Now we have seen, in general, that the line joining any point on the curve aẞy to another point a'ß'y' meets the curve again in a point, whose coordinates are S'a - 2P'a', S'ẞ- 2P'ß', S'y - 2P'y. Now if the point a'ß'y' be the intersection of lines ẞ, Y, we may take a' = 1, ß′ = 0, y′ = 0, which gives S'a, P' S1, and the coordinates of the point where the line joining aßy to ẞy meets the curve, are aa - 281, aß, ay. In like manner, the line joining aßy to ya, meets the curve again in ba, bß – 282, by. The line joining these two points will pass through aß, if which is the condition to be fulfilled by the coordinates of the vertex. Writing in this equation aa = S1 - hẞ - gy, bẞ = S2 - ha -fy, it becomes h (aS1 + ẞS2) + y (ƒS1 + gS2) = 0. But since aẞy is on the curve, aS1 + BS2 + yS, = 0, and the equation last written reduces to y (ƒS1+ gs2- hSg) = 0. Now the factory may be set aside as irrelevant to the geometric solution of the problem; for although either of the points where y meets the curve fulfils the condition which we have expressed analytically, namely, that if it be joined to ẞy and to ya, the joining lines meet the curve again in points which lie on a line with aß; yet, since these joining lines coincide, they cannot be sides of a triangle. The vertex of the sought triangle is therefore either of the points where the curve is met by ƒS1+gS2-hS3. It can be verified immediately that ƒS, = gS2 = hS, denote the lines joining the corresponding vertices of the triangles aßy, SS2S3. Consequently (see Ex. 2, Art. 60), the line ƒS, +gS2- hS, is constructed as follows: "Form the triangle DEF whose sides are the polars of the given points A, B, C; let the lines joining the corresponding vertices of the two triangles meet the opposite sides of the polar triangle in L, M, M; then the lines LM, MN, NL pass through the vertices of the required triangles." L M E N The truth of this construction is easily shown geometrically: for if we suppose that we have drawn the two triangles 123, 456 which can be drawn through the points A, B, C; then applying Pascal's theorem to the hexagon 123456, we see that the line BC passes through the intersection of 16, 34. But this latter point is the pole of AL (Ex. 1, Art. 146). Conversely, then, AL passes through the pole of BC, and L is on the polar of A (Ex. 1, Art. 146). This construction becomes indeterminate if the triangle is selfconjugate in which case the problem admits of an infinity of solutions. N N. Ex. 8. If two conics have double contact, any tangent to the one is cut har monically at its point of contact, the points where it meets the other, and where it meets the chord of contact. If in the equation S+ R2 = 0, we substitute la'+ma", lẞ′ + mß", ly' +my", for a, ß, y, (where the points a'ß'y', a"B"y" satisfy the equation S = 0), we get (IR' + mR")2 + 2lm P = 0. Now, if the line joining a'ß'y', a"B"y", touch S+ R2, this equation must be a perfect square; and it is evident that the only way this can happen is if P = − 2R'R", when the equation becomes (IR' — mR'')2 = 0; when the truth of the theorem is manifest. Ex. 9. Find the equation of the conic touching five lines, viz. a, ß, y, Aa+Bß+Cy, A'a + B'ß + Cy. Ans. (la)* + (mß)* + (ny)*, where l, m, n are determined by the conditions Ex. 10. Find the equation of the conic touching the five lines, a, ß, y, a + ß + y, 2a + B-Y. We have l+m + n = 0, 1 l + m − n = 0: hence the required equation is 2 (− a)* + (3ß3)* + (y)* = 0. Ex. 11. Find the equation of the conic touching a, ß, y, at their middle points. Ans. (aa) + (3B) + (y)=0 Ex. 12. Find the condition that (la) + (mp)* + (ny)* = 0 should represent a para Ex. 13. To find the locus of the focus of a parabola touching a, ß, y. Generally, if the coordinates of one focus of a conic inscribed in the triangle aßy be a'ß'y', the lines joining it to the vertices of the triangle will be and since the lines to the other focus make equal angles with the sides of the triangle (Art. 189), these lines will be (Art. 55) Hence, if we are given the equation of any locus described by one focus, we can at once write down the equation of the locus described by the other; and if the second focus be at infinity, that is, if a" sin A + ẞ" sin B + y" sin C = 0, the first Ex. 12) these values satisfy both the equations, a sin A + ẞ siu B + y sin C = 0, Jla + Jmß + Jny = 0. sin2A sin2B sin2C The coordinates, then, of the finite focus are m n |