FM; if we suppose it measured from the side FV, the equation and is, therefore, one of a class of equations, p" cos no = a", some of whose properties we shall mention hereafter. CHAPTER XIII. EXAMPLES AND MISCELLANEOUS PROPERTIES OF THE CONIC SECTIONS. 230. THE method of applying algebra to problems relating to conic sections is essentially the same as that employed in the case of the right line and circle, and will present no difficulty to any reader who has carefully worked out the Examples given in Chapters III. and VII. We, therefore, only think it necessary to select a few out of the great multitude of examples which lead to loci of the second order, and we shall then add some properties of conic sections, which it was not found convenient to insert in the preceding chapters. Ex. 1. A line of constant length moves about in the legs of a given angle: to find the locus described by a fixed point on it. Denoting PL by n, PK by m, and LK by 1, we have, by similar triangles, L N P the equation of an ellipse having the point O for its centre, since B 4AC is here ne Ex. 2. If P be a fixed point, and LK any right line drawn through it, to find the locus of intersection of the parallels to OK, OL, through the points L and K. Ex. 3. Or of perpendiculars erected to OK, OL, through the same points. Ex. 5. Two equal rulers, AB, BC, are connected by a pivot at B; the extremity A is fixed, while the extremity C is made to traverse the right line AC; find the locus described by any fixed point P on BC. Ex. 6. Given base and difference of base angles of a A triangle: to find the locus of vertex. B O P We may proceed exactly as at page 85, where the sum of the base angles is given. The locus will be found to be an equilateral hyperbola, of which the base is a diameter. The difference of base angles being given, it is easy to see that the internal and external bisectors of the vertical angle must be parallel to fixed lines, and these lines will be parallel to the asymptotes of the locus. Conversely, if we consider the triangle whose base is any diameter of an equilateral hyperbola, and whose vertex is on the curve, the sides are parallel to conjugate diameters (Art. 183); but conjugate diameters of an equilateral hyperbola make equal angles with the asymptotes (Art. 178). Ex. 7. Given base and the product of the tangents of the base angles of a triangle: find the locus of vertex. It will be a conic section, of which the extremities of the base are vertices. This is the converse of Art. 174. Ex. 8. Given base and the product of the tangents of the halves of the base angles : find the locus of vertex. Expressing the tangents of the half angles in terms of the sides, it will be found that the sum of sides is given: and, therefore, that the locus is an ellipse, of which the extremities of the base are the foci. Ex. 9. Given base and sum of sides of a triangle: find the locus of the centre of the inscribed circle. It may be immediately inferred, from the last two examples, that the locus is an ellipse, whose vertices are the extremities of the given base. Ex. 10. Given the vertical angle of a triangle in magnitude and position, and also the area: find the locus of a point dividing the base in a given ratio. Ex. 11. Given base of a triangle, and that one base angle is double the other; find locus of vertex. Ex. 12. Trisect a given arch of a circle. Ans. The point of trisection is determined as the intersection of the given arch with a given hyperbola. Ex. 13. Given base and area of a triangle; find the locus of the intersection of perpendiculars. Ex. 14. Find the locus of the centre of a circle which touches two others; or which touches a given circle and a given right line. Ex. 15. Given the base of a triangle, and the length of the intercept made by the sides on a given line; find the locus of vertex. Ex. 16. Two vertices of a given triangle move along fixed right lines; find the locus of the third. Ex. 17. Two vertices of a triangle move along fixed right lines, and the sides pass through fixed points; find the locus of the third vertex. Ex. 18. Find the locus of the centre of a circle which makes given intercepts on two given lines. Ex. 19. A triangle ABC circumscribes a given circle; the angle at C is given, and B moves along a fixed line; find the locus of A. Let us use polar co-ordinates, the centre O being the pole, and the angles being measured from the perpendicular on the fixed line; let the co-ordinates of A, B, be p, 0; p', 0'. Then we have p' cos 0' = p. But it is easy to see that the angle AOB is given (= a). And since the perpendicular of the triangle AOB is given, we have But 0+0= a; therefore the polar equation of the locus is r2 = which represents a conic. p2 p2 sin2a p2 cos2 (a−0) + p2 – 2pp cos a cos (a – 0)' Ex. 20. Given two conic sections, to find the locus of the pole, with respect to one, of any tangent to the other. Ax2+Bxy+Cy2 + Dx + Ey + F = 0, the polar of any point, with regard to the second, is (Art. 144) (2Ax' + By + D) x + (2Cy' + Bx' + E) y + Dx' + Ey' + 2F = 0. But the condition that this should touch the first is (p. 150) a2 (2Ax' + By + D)2 + b2 (2Cy' + Bx' + E)2 = (Dx' + Ey' + 2F)2. This condition, which must be satisfied by the point (x'y'), is the equation of its locus, and is plainly of the second degree. 231. We give in this Article some examples on the focal properties of conics. Ex. 1. The distance of any point on a conic from the focus is equal to the whole length of the ordinate at that point, produced to meet the tangent at the extremity of the focal ordinate. Ex. 2. If from the focus a line be drawn making a given angle with any tangent, find the locus of the point where it meets it. Ex. 3. To find the locus of the pole of a fixed line with regard to a series of concentric and confocal conic sections. Now, if the foci of the conic are given, a2 – b2 = c2 is given; hence, the locus of the pole of the fixed line is mx ny = c2, the equation of a right line perpendicular to the given line. If the given line touch one of the conics, its pole will be the point of contact (Art. 144). Hence, given two confocal conics, if we draw any tangent to one and tangents to the second where this line meets it, these tangents will intersect on the normal to the first conic. Ex. 4. The focus being the pole, prove that the polar equation of the tangent, at the Р 2p point whose angular co-ordinate is a, is - = e cos 0 + cos(0 − a). This expression is due to Mr. Davies (Philosophical Magazine for 1842, p. 192, cited by Walton, Examples, p. 368). Ex. 5. Prove that the polar equation of the chord through points whose angular coordinates are a + ẞ, a - ẞ, is This expression is due to Mr. Frost (Cambridge and Dublin Math. Journal, i. 68, cited by Walton, Examples, p. 375). These equations may be conveniently used in investigating theorems concerning angles subtended at the focus. Still simpler methods, however, of obtaining these will be given in Chapter XIV. Ex. 6. If a chord PP' of a conic pass through a fixed point O, then tan PFO. tan P'FO is constant. The reader will find an investigation of this theorem by the help of the equation of the last Example (Walton's Examples, p. 377). I insert here the geometrical proof given by Mr. MacCullagh, to whom, I believe, the theorem is due. Imagine a point O taken anywhere on PP' (see figure, p. 187), and let the distance FO be e' times the distance of O from the directrix; then since the distances of P and O from the directrix are proportional to PD and OD, we have or, since (Art. 196) PFT is half the sum, and OFT half the difference, of PFO and P'FO It is obvious that the product of these tangents remains constant if O be not fixed, but be anywhere on a conic having the same focus and directrix as the given conic. Ex. 7. If normals be drawn at the extremities of any focal chord, a line drawn through their intersection parallel to the axis will bisect the chord. a2 Take any point on the directrix whose co-ordinates are x = y=ẞ, then the equa с tion of the polar of that point, which passes through the focus, will be Substituting for x from this equation in the equation of the curve, the ordinates of the points where this line meets the curve are given by the equation Hence, if y', y", be the ordinates of the point of intersection, we have It may be found, in like manner, that the abscissæ of the intersection of the chord with the curve are determined by the equation Ex. 8. If a chord pass through a focus, the line joining the intersection of tangents at its extremities to the intersection of the corresponding normals will pass through the other focus. The equation of the joining line is cẞ (x + c) = (a2 + c2) y. Ex. 9. Find the locus of the intersection of normals at the extremities of a focal chord. or (a2 + c2)2y2 = b2 (c + x) (c3 − u2x), (a2 + c2)2 y2 + a2b2x2 + b1 cx = b2c1. Ex. 10. If 0 be the angle between the tangents to an ellipse from any point P; and if p, p' be the distances of that point from the focus, prove that cos 0= 02+p22-4a2 |