3. a negative and b positive; :: y = ∞ x+b; join DE; DE produced is the required locus. 4. a negative and b negative; Let x = 0; :: y = y = 0; .. y= απ b; b; in A take A Cb; join BC; BC produced is the required locus. b ; α 35. The quantities a and b may also change in absolute value. Let b = 0; y = ±a x; and the loci are two straight lines passing through the origin and drawn at angles with the axis of x whose respective tangents are a. Let a = 0; y = 0x±b; :.y = ±b and x = these results shows that every point in the locus is equidistant from the axis of x, and the latter (or 0x0) that every value of x satisfies the original equation; hence the loci are two straight lines drawn through D and C both parallel to the axis of x. It has been stated (28), that the system of equations yb, x = 0 refers to a point; we here see that the system y=b, x = straight line; hence, although the equation x = is generally omitted, yet it must be considered as essential to the locus. ; referring to article 32, the equation to the straight line 0 y may be written yaxa a or xa, which when a = α becomes 0 y = xa; hence, as before, the system xa, y = or more simply the equation x = the axis of y and at a distance a denotes two straight lines parallel to a from that axis. Again let both a = 0, and b = 0; and .. the equation y = ax + b becomes y 0 x + 0; and hence, y = 0, x = and y = 0 and the locus is and 60, the equation becomes 0 y = + 0 ; :. x = 0 Hence the equation x = 0 denotes the axis of y. 36. By the above methods the line to which any equation of the first order belongs may be drawn. In the following examples reference is made to parts of the last figure. 5x 1= 0; let x = 0, .. y= 1 3 Ex. 1. 3 y ; on the axis AY take A D one-third of the linear unit, then the line passes through D: 1 5 again let y = 0, .. x=- ; on the axis Ax take A B = of the unit, 5 then the line passes through B; hence the line joining the points B and D is the locus required. 21x+60; a line situated like C E. Ex. 3. y x = 0; let x = 0.. y = 0, and the line passes through the origin; also a or the tangent of the angle which the line makes with the axis of x 1, therefore that angle 45°; hence the straight line drawn through the origin and bisecting the angle Y AX is the required locus. Ex. 4. 5 y 2 x 0. The line passes through the origin as in the last example, but to find another point through which the line passes, let x = 5; :. y = 2: hence take AE 5, and from E draw EP (= 2) perpendicular to AX; then the line joining the points A, P is the locus required. Ex. 5. ay + bx = 0; a line drawn through A, and parallel to B C. Ex. 6. y2 3x2= 0; two straight lines making angles of 60° with the axis of x. 20; take AE = 1, and A B = 2, the lines drawn drawn parallel to A X is the locus. Ex. 8. x2 + x through E and B parallel to A Y are the required loci. Ex. 9. y + 2x = 4. The equation to a straight line may be put 1, and since when y = 0, x = a, y Ꮖ under the convenient form + b α and when x = 0, y = b, the quantities b and a are respectively the distances of the origin from the intersection of the line with the axes of y and x. Thus Ex. 9. in this form is + = 1, take A D = 4, and A E = 2, join D E, this line produced is the required locus. 37. If the equation involve the second root of a negative quantity its locus will not be a straight line, but either a point or altogether imaginary : thus the locus of the equation y + 2 x √ — 1 — a = 0 is a point whose co-ordinates are x = 0 and y = a, for no other real value of x can give a real value to y; but the locus of the equation y + x + a √ − 1 = 0 is imaginary, for there are no corresponding real values of x and y. (24) 38. We have thus seen that the equation to a straight line is of the form yax + b, and that its position depends entirely upon a and b. By a given line we understand one whose position is given, that is, that a and b are given quantities; when we seek a line we require its position, so that assuming y= ax + b to be its equation, a and b are the two indeterminate quantities to be found by the conditions of the question: if only one can be found the conditions are insufficient to fix the position of the line. By a given point we understand one whose co-ordinates are given; we shall generally use the letters x, and y, for the co-ordinates of a given point, and to avoid useless repetition, the point whose co-ordinates are x1 and y1 will be called "the point 1, Y1." Similarly the line whose equation is y = ax + b will be called the line y ax + b.” If in the same problem we use the equations to two straight lines as y = ax + b and y ax b', it must be carefully remembered that a and y are not the same quantities in both equations; we might have used the equations y = ax + b, and Ya' X + b', X and Y being the variable coordinates of the second line, but the former notation is found to be the more convenient. 39. We regret much that in the following problems on straight lines we cannot employ an homogeneous equation as y X b a 1. In algebraical geometry the formulas most in use are very simple, much more so indeed than they would be if homogeneous moreover the advantage of a uniform system of symbols and formulas is so great to mathematicians that it should not be violated without very strong reasons. To remedy in some degree this want of regularity, the student should repeatedly consider the meaning of the constants at his first introduction to the subject of straight lines. PROBLEMS ON STRAIGHT LINES. 40. To find the equation to a straight line passing through a given point. The point being given its co-ordinates are known; let them be x1 y1, and let the equation to the straight line be y = ax + b; we signify that this line passes through the point x, y, by saying that when the variable abscissa x becomes x, then y will become y1: hence the equation to the line becomes substituting this value for b in the first equation, we have The shortest method of eliminating b is by subtracting the second equation from the first, and this is the method generally adopted. Since a, which fixes the direction of the line, is not determined, there may be an infinite number of straight lines drawn through a given point; this is also geometrically apparent. If the given point be on the axis of x, y1 = 0, and .. y = a (x − x1); and if it be on the axis of y, x1 = 0 :. y — y1 = α x. If either or both of the co-ordinates of the given point be negative, the proper substitutions must be made: thus if the point be on the axis of t and in the negative direction from A, its co-ordinates will be therefore the equation to the line passing through that point will be y = a (x + x1). x1 and 0; 41. To find the equation to a straight line passing through two given points x, y, and X, Y2 Let the required equation be y=ax+b (1) then since the line passes through the given points, we have the equations X1 X2 Substituting this value of a in (4), we have finally The two conditions have sufficed to determine a and b, and by their elimination the position of the line is fixed, as it ought to be, since only one straight line can be drawn through the same two points. This equation will assume different forms according to the particular situation of the given points. Thus if the point 2, y2 be on the axis of r, we have y = 0; This last equation is also thus obtained; the line passing through the origin, its equation must be of the form y=ax (31) where « is the tangent of the angle which the line makes with the axis of x, and this line passing through the point x, y, a must be equal to- :.y= x. Yi If a straight line pass through three given points, the following relation must exist between the co-ordinates of those points: (Y1 X-X1 Y2)-(Y1 X3 — X1 Y3)+(Y2 X3 — X2 Y3) = 0. 42. To find the equation to a straight line passing through a given point x, y, and bisecting a finite portion of a given straight line. Let the portion of the straight line be limited by the points x, y, and x1 + x2 x2 y2, and therefore the co-ordinates of the bisecting point are 2 43. To find the equation to a straight line parallel to a given straight then since the lines are parallel they must make equal angles with the axis of x or a' « .. the required equation is y= ax + b' (3). Of course b' could not be determined by the single condition of the parallelism of the lines, since an infinite number of lines may be drawn parallel to the given line; but if another condition is added, b' will be then determined: thus if the required line passes also through a given point x, y, equation (2) is 44. To find the intersection of two given straight lines C B, ED. This consists in finding the co-ordinates of the point O of intersection. Now it is evident that at this point they have the same abscissa and ordinate; hence if in the equations to two lines we regard as representing the same abscissa and y the same ordinate, it is in fact saying that they are the co-ordinates X, Y of the point of intersection O. |