19. Find the locus of the ultimate intersections of a system of lines defined by the equation 20. y cos-x sin =c-c sin log tan where is the variable parameter. + Result. 2y=c(e+e ̃3). = The equation to a spiral is " cos no a"; if perpendiculars are drawn through the extremities of the radii vectores the envelop of the perpendiculars is the 21. A series of ellipses has the same centre and directrix; shew that the envelop is a pair of parabolas, but that the envelop will not meet those ellipses whose excentricity is less than 1 22. Find the locus of the ultimate intersections of an ellipse which touches a given line at a given point at the extremity of the axis minor, the excentricity varying as the axis major. What are the limits of the excentricity in order that two consecutive ellipses may intersect? 23. A line is drawn from the focus to any point of a conic section, and a circle is described upon it as a diameter; shew that the locus of the ultimate intersections of all such circles is a circle, except, in a certain case, where it is a right line. 24. Shew that the locus of the ultimate intersections of all the chords of an ellipse which join the points of contact of pairs of tangents at right angles to one another is a confocal ellipse. 25. Find the locus of the ultimate intersections of the lines x cos 30+ y sin 30= a cos 20), where is the variable parameter. Result. (x2 + y2)2 = a2 (x2 — y2). 26. Find the envelop of the circles described on the radii of an ellipse, drawn from the centre, as diameters. Result. (x2+ y2)2 = a2x2 +b2y2. 0 27. On radius vector of the curve r = c sec" as diameter n any is described a circle; shew that the envelop of all such circles is the curve rc sec" Ꮎ n 1° 28. Find the locus of the ultimate intersections of a family of parabolas of which the pole of a given equiangular spiral is the focus, and its tangents directrices. Result. A similar equiangular spiral. 29. Perpendiculars are drawn from the pole of an equiangular spiral on the tangents to the curve; find the envelop of the circles described on these perpendiculars as diameters. Result. A similar equiangular spiral. 30. From every point of a parabola as centre a circle is described with a radius exceeding the focal distance of the point by a constant quantity; find the envelop of the circles. Result. (x+c+ a) {y2 + (x − a)2 — c2)=0; where c is the constant quantity. 31. Find the envelop of the straight lines ax sec - by cosec 0 = a2 — b2. Result. (ax)3 + (by)3 — (a2 — b2)3. = 32. From a fixed point A in the circumference of a circle any chord AP is drawn and bisected in H, and on PH as diameter a circle is described; find the locus of the ultimate intersections of the system of circles described according to this law. Result. a2 (x2+y)= (2x2 + 2y — 3ax)2; where 2+ y2=2ax is the equation to the given circle. CHAPTER XXVI. TRACING OF CURVES. 340. In this chapter we shall give some examples of tracing curves from their equations. First find the value of dy; taking the logarithms of both sides of the equation and differentiating, we have 1 dy y dx х x √(x2 - 4a2) (1 dx √(x2 - a2) 4a2 2 Also when xa we see that y is infinite. We may now assign different values to x, and note the dy corresponding values of y and obtained from (1) and (2). dx Since the curve is symmetrical with respect to the axis of x, we may confine our attention to the positive values of y. When x is greater than 2a, y is possible. It is not necessary to give negative values to x in this example, because the curve is symmetrical with respect to the axis of y. If we draw the asymptotes and make use of the above list of particular values of y and dy, we shall have sufficient d'y dx2, in order to dy materials for ascertaining the general form of the curve. If necessary, in any example, we may find determine the points of inflexion; also by examining when dx vanishes, we can determine the maxima and minima values of y. If we take the upper sign in equation (3), we have for the asymptote y=x (4); (5). When x is very large the terms included in the &c. of equation (5) will be very small compared with 3a2 Hence comparing (4) and (5) we see that corresponding to the same abscissa the ordinate of the curve is less than that of the asymptote, and therefore the curve lies below the asymptote as represented in the figure. |