Now, if the foci of the conic are given, a2 — b2 = e2 is given; hence, the locus of the pole of the fixed line is 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. 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. Find the locus of the points of contact of tangents to a series of confocal ellipses from a fixed point on the axis major. Ans. A circle. Ex. 5. The lines joining each focus to the foot of the perpendicular from the other focus on any tangent intersect on the corresponding normal and bisect it. Ex. 6. The focus being the pole, prove that the polar equation of the chord through points whose angular coordinates are a + ß, a − ß, is Р This expression is due to Mr. Frost (Cambridge and Dublin Math. Journal, 1., 68, cited by Walton, Examples, p. 375). It follows easily from Ex. 3, p. 37. Ex. 7. The focus being the pole, prove that the polar equation of the tangent, at Р the point whose angular coordinate is α, is e cos 0+ cos (0 - a). 2p This expression is due to Mr. Davies (Philosophical Magazine for 1842, p. 192, cited by Walton, Examples, p. 368). Ex. 8. If a chord PP' of a conic pass through a fixed point 0, then is constant. tan PFO.tan P' FO The reader will find an investigation of this theorem by the help of the equation of Ex. 6 (Walton's Examples, p. 377). I insert here the geometrical proof given by Mr. Mac Cullagh, to whom, I believe, the theorem is due. Imagine a point O taken anywhere on PP' (see figure p. 206), 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. 191) PFT is half the sum, andOFT 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. 9. To express the condition that the chord joining two points x'y', x"y" on the curve passes through the focus. = This condition may be expressed in several useful of which are got by expressing that 0" made with the axis by the lines joining the sin 0"=- sin e' gives equivalent forms, two of the most 0'+ 180°, where e', 0" are the angles focus to the points. The condition 0; a (y' + y′′) = e (x'y" + x"y'). Ex. 10. If normals be drawn at the extremities of any focal chord, a line drawn through their intersection parallel to the axis major will bisect the chord. [This solution is by Larrose, Nouvelles Annales, XIX. 85.] Since each normal bisects the angle between the focal radii, the intersection of normals at the extremities of a focal chord is the centre of the circle inscribed in the triangle whose base is that chord, and sides the lines joining its extremities to the other focus. Now if a, b, c be the sides of a triangle whose vertices are x'y', x'y', x'"'y"", then, Ex. 6, p. 6, the coordinates of the centre of the inscribed circle are In the present case the coordinates of the vertices are x', y'; x", y"; — c, 0; and the lengths of opposite sides are a + ex', a + ex′, 2a - ex′ — ex". We have therefore (a + ex′) y" + (a + ex′′) y' y= 4a or, reducing by the first relation of the last Example, y = § (y' + y′′), which proves the theorem. We could find, similarly, expressions for the coordinates of the intersection of tangents at the extremities of a focal chord, since this point is the centre of the circle exscribed to the base of the triangle just considered. The line joining the intersection of tangents to the corresponding intersection of normals evidently passes through a focus, being the bisector of the vertical angle of the same triangle. Ex. 11. To find the locus of the intersection of normals at the extremities of a focal chord. Let a, ẞ be the coordinates of the middle point of the chord, and we have, by the last Example, a2 (x + c) '; B = 1 (y' + y') = y. If, then, we knew the equation of the locus described by aß, we should, by making the above substitutions, have the equation of the locus described by xy. Now the polar equation of the locus of middle point, the focus being origin, is (Art. 193) which, transformed to rectangular axes, the centre being origin, becomes b2a2 + a2ß2 = b2ca. The equation of the locus sought is, therefore, a2b2 (x + c)2 + (a2 + c2)2y2 = b2c (a2 + c2) (x + c). Ex. 12. If @ 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 P foci, prove that cos 0 = p2 + p22 - 4a2 T' (see also Ex. 13. If from any point O two lines be drawn to the foci (or touching any confocal conic) meeting the conic in R, R'; S, S'; then (see also Ex. 15, Art. 231) 1 = 1 1 1 OS' [Mr. M. Roberts.] It appears from the quadratic, by which the radius vector is determined (Art. 136), that the difference of the reciprocals of the roots will be the same for two values of 0, which give the same value to (ac-g2) cos2 0 + 2 (ch — gf) cos 0 sin 0 + (bc −ƒ2) sin2 0. Now it is easy to see that A cos20 + 2H cos 0 sin 0 + B sin20 has equal values for any two values of 0, which correspond to the directions of lines equally inclined to the two represented by Ax2 + 2Hxy + By2 = 0. But the function we are considering becomes = 0 for the direction of the two tangents through O (Art. 147); and tangents to any confocal are equally inclined to these tangents (Art. 189). It follows from this example that chords which touch a confocal conic are proportional to the squares of the parallel diameters (see Ex. 15, Art. 231). 227. We give in this Article some examples on the parabola. The reader will have no difficulty in distinguishing those of the examples of the last Article, the proofs of which apply equally to the parabola. Ex. 1. Find the coordinates of the intersection of the two tangents at the points x'y', x'y', to the parabola y2 = px. y' Ans. y=+y", x= 2 y'y' p Ex. 2. Find the locus of the intersection of the perpendicular from focus on tangent with the radius vector from vertex to the point of contact. Ex. 3. The three perpendiculars of the triangle formed by three tangents intersect on the directrix (Steiner, Gergonne, Annales, XIX. 59; Walton, p. 119). The equation of one of those perpendiculars is (Art. 32) The symmetry of the equation shows that the three perpendiculars intersect on the directrix at a height 2y'y"y"", y'+y" + y"" y= + D2 2 Ex. 4. The area of the triangle formed by three tangents is half that of the triangle formed by joining their points of contact (Gregory, Cambridge Journal, 11. 16 Walton, p. 137. See also Lessons on Higher Algebra, Ex. 12, p. 15). Substituting the coordinates of the vertices of the triangles in the expression of Art. 36, we find for the latter area (y′ — y′′) (y′′ — y''') (y'"'—y'); and for the former 2p area half this quantity. Ex. 5. Find an expression for the radius of the circle circumscribing a triangle inscribed in a parabola. Σ is easily proved to be def. But if d be the length of the 4Σ The radius of the circle circumscribing a triangle, the lengths of whose sides are d, e, f, and whose area = chord joining the points x"y", x""'y"", and e' the angle which this chord makes with the axis, it is obvious that d sin e'=y" -y"". Using, then, the expression for the area found in the last Example, we have R = We might express the radius, also, in terms of the focal chords parallel to the sides of the triangle. For (Art. 193, Ex. 2) the length of a chord making an angle 0 with the axis c'c'c'" Hence R2 = 4p is c = Ρ sin20 Ρ It follows from Art. 212 that c', c", c"" are the parameters of the diameters which bisect the sides of the triangle. Ex. 6. Express the radius of the circle circumscribing the triangle formed by three tangents to a parabola in terms of the angles which they make with the axis. p 8 sin e' sin e" sin "" 9 64p Ans. R ; or R2- =P'p"p"" where p', p", p"" are the parameters of the diameters through the points of contact of the tangents (see Art. 212). Ex. 7. Find the angle contained by the two tangents through the point x'y' to the parabola y2 = 4mx. The equation of the pair of tangents is (as in Art. 92) found to be (3'2 — 4mx') (y2 - 4mx) = {yy' — 2m (x + x')}2. Ex. 8. Find the locus of intersection of tangents to a parabola which cut at a given angle. Ans. The hyperbola, y2 - 4mx = (x + m)2 tan2p, or y2 + (x — m)2 = (x + m)2 sec2p. From the latter form of the equation it is evident (see Art. 186) that the hyperbola has the same focus and directrix as the parabola, and that its eccentricity = sec p. Ex. 9. Find the locus of the foot of the perpendicular from the focus of a parabola on the normal. The length of the perpendicular from (m, 0) on 2m (y — y') + y' (x − x') = 0 is But if 0 be the angle made with the axis by the perpendicular (Art. 212) Ex. 10. Find the coordinates of the intersection of the normals at the points x'y', x"y". y'y" (y'+y") 4m ,y= 8m2 Or if a, ẞ be the coordinates of the corresponding intersection of tangents, then (Ex. 1) x = 2m+ B2 m m Ex. 11. Find the coordinates of the points on the curve, the normals at which pass through a given point x'y'. Solving between the equation of the normal and that of the curve, we find 2y3 + (p2 - 2px') y = p2y', and the three roots are connected by the relation y1+Y2+Y3 = 0. The geometric meaning of this is, that the chord joining any two, and the line joining the third to the vertex, make equal angles with the axis. Ex. 12. Find the locus of the intersection of normals at the extremities of chords which pass through a given point x'y'. We have then the relation ẞy' = 2m (x' + a); and on substituting in the results of Ex. 10 the value of a derived from this relation we have 2mx+By' = 4m2 + 2ẞ2 + 2mx'; 2m2y = 2ẞmx' — ẞ2y' ; whence, eliminating ß, we find 2 {2m (y − y') + y′ (x − x′)}2 = (4mx′ — y′2) (y'y + 2x′x — 4mx′ — 2x′2), the equation of a parabola whose axis is perpendicular to the polar of the given point. If the chords be parallel to a fixed line, the locus reduces to a right line, as is also evident from Ex. 11. Ex. 13. Find the locus of the intersection of normals at right angles to each other. In this case a =m, x = 3m + Ex. 14. If the lengths of two tangents be a, b, and the angle between them w, find the parameter. Draw the diameter bisecting the chord of contact; then the parameter of that diameter is p' = (where is the length of the perpendicular on the chord from the intersection of the tangents). But 2y=ab sin w, and 16x2 = a2 + b2 + 2ab cos w ; hence Ex. 15. Show, from the equation of the circle circumscribing three tangents to a parabola, that it passes through the focus. The equation of the circle circumscribing a triangle being (Art. 124) By sin A+ ya sin B + aß sinC = 0; the absolute term in this equation is found (by writing at full length for a, x cos a + y sin a-p, &c.) to be p'p" sin (ẞ- y) + p′′p sin (y — a) + pp' sin (a - ẞ). But if the line a be a tangent to a parabola, and the origin the focus, we have (Art. 219) m and the absolute term |