1832 1833 36. When a series of algebraical quantities is to be represented in one line, and each of them measured from the same point, the positive quantities being represented by lines taken in one direction, the negative quantities must be represented by lines taken in the opposite direction. 37. Shew how the cycloid is traced out, and find its equation. 38. Having given the coordinates to a point and the equation to a straight line, find the distance of a point from the straight line. 39. Find the equation to a straight line which passes through a given point and cuts the axis of x at an angle of 135o. 40. Every curve of an odd degree either in x or y has at least one infinite branch. 41. Express the sine of the angle which a straight line makes with a plane in terms of the coefficients of their equations, and hence determine the relation between the coefficients when the straight line is perpendicular to the plane. 42. Define the centre of a curve, and shew how it may be determined. In what cases has the curve defined by the general equation of the second order a centre ? 43. Find the equation to a straight line : determine the distances from the origin at which it cuts the axes of x and y, and the cosine of the angle which it makes with a given straight line passing through the origin. 44. If an odd number of equidistant ordinates be drawn in a curve, the area between the extreme ordinates may be found approximately by adding the first and last ordinates to four times the sum of the even ordinates, and twice the sum of the odd ones, and multiplying by į their common distance. 45. Find the equation to a straight line drawn through a given point in the axis of y, and making an angle of 60° with a given straight line. 46. In rolling a circular board on level ground at the foo of a wall, in such a manner that the plane of the board was 1834 1836 parallel to the wall, a nail projecting out of the wall traced a curve upon the board : required its equation. Deduce a simple method of describing the spiral of Archimedes by a continuous motion. 47. Prove the following method of drawing a tangent to any curve of the second order from a given point P without it. From P draw any two lines, each cutting the curve in two points. Join the points of intersection two and two, and let the points in which the joining lines (produced if necessary) cross each other be joined by a line which will, in general, cut the curve in two points A, B. PA, PB are tangents at A and B. 48. Investigate formulæ for transforming the equation of a curve from one system of oblique coordinate axes to another having the same origin. Transform the equation xy = c?, where the axes are inclined to each other at a given angle, to one where the axes bisect the angles made by the former. 49. Define a cycloid and investigate its equation. Prove that the length of an arc measured from the vertex is equal to twice the corresponding chord of the generating circle. SECTION VII. QUESTIONS IN CONIC SECTIONS. 1821 1822 1. Of all sections made by planes passing through both sides of an oblique cone, two are circles, and all the rest ellipses. 2. S being the focus of an hyperbola, and (pm) the perpendicular upon its directrix from a point (p) in the opposite hyperbola, Sp: pm :: SC: AC, a given ratio. 3. Two tangents to a parabola drawn from the same point of the directrix, are at right angles to each other. 4. PSp is any parameter of a parabola whose focus is S and latus rectum L, prove that 4SP. Sp = L (SP + Sp). 5. The ordinate to the axis of an ellipse is produced till it equals the corresponding subtangent: find the equation to the curve thus traced out, and its area. 6. Putting A and B for the sectors CAP, CAp of a rectangular hyperbola, whose semi-axis CA = 1, and calling the abscissas CN, Cn the cosines, and the ordinates PN, pn the sines of A and B, then will sin (A + B) sin A cos B + cos A sin B, and cos (A + B) = cos A cos B sin A sin B. 7. If in an ellipse there be taken three abscissas in arithmetic progression, the radius vectors drawn from the focus to the extremities of the ordinates at those points will also be in arithmetic progression. 1823 8. If tangents drawn to any two points of an ellipse meet each other; shew that their lengths are inversely as the sines of the angles which they make with the lines drawn to either focus. 9. A triangle is described about an ellipse : prove that the products of the alternate segments of the sides, made by the points of contact, are equal. 10. If two equal parabolas have a common axis, a straight 1824 line touching the interior and bounded by the exterior will be bisected in the point of contact. 11. If two lines revolving in the same plane round the points 12. If a right cone whose vertical angle is 90°, be cut by y2 – 2xy + 22 8x + 16 = 0. 15. From the vertex of a parabola a straight line is dr 16. In a parabola, in the focal distance SP, Sp is taken 1825 equal to the ordinate PN. Find the equation to the curve traced out by the point p. 17. The centre of an ellipse coincides with the vertex of common parabola, and the axis major of the ellipse is perpendicular to the axis of the parabola. Required the proportion of the axes of the ellipse, so that it may cut the parabola at right angles. a 1826 COS 2A + 18. Having given two conjugate diameters of an ellipse and the angle contained between them; find the magnitudes and positions of the axes. 19. The abscissa and double ordinate of a common parabola are a and b, and the diameters of its circumscribed and inscribed circles D and d; prove that D + d = a + b. 20. Three straight lines revolve about three given points not in the same straight line, and intersect one another in three points; prove that if the loci of two of these intersections be straight lines, the locus of the third will be a conic section. 21. If r be the radius vector of an ellipse from the centre, and @ the angle which it makes with the major axis; it is required to express r in a series of the form A. + A, cos 0 + A, and to shew particularly how A., Ay, and A, may be determined. 22. Through any point in the straight line joining the centre and intersection of the tangents to any two points, of an ellipse, two straight lines are drawn respectively parallel to its diameters passing through the points of contact; prove that the triangles formed by these lines and the tangents are equal. 23. Shew that the parameter belonging to any diameter of a parabola varies inversely as the square of the sign of the angle at which the corresponding ordinates are inclined to it. 24. If aya + bxy + cx? + dy + ex + f 0, be the equation to a curve of the second order; prove that the angles which its principal diameters make with the axis of x, may be determined from the equation b tan 20 c a 1827 25. If C be the centre of an ellipse, and in the normal to any point P, PQ be taken equal to the semi-conjugate at P, Q will trace out a circle round C. 26. Given the radius vector at any point of a parabola, and the angle it makes with the curve; find the latus rectum and the place of the vertex. |