CHAPTER II. 13. THE STRAIGHT LINE. To find the equation of a straight line parallel to one of the co-ordinate axes. Let LP be a straight line parallel to the axis of x and meeting the axis of y at L, and let OL=b. Let x, y be the co-ordinates of any point P on the line. Hence y=b is the equation of the line. Similarly a is the equation of a straight line parallel to the axis of y and at a distance a from it. 14. To find the equation of a straight line which passes through the origin. Let OP be a straight line through the origin, and let the tangent of the angle XOP=m. Let x, y be the co-ordinates of any point P on the line. Then Hence NP=tan NOP. ON. y=mx is the required equation. N 15. To find the equation of any straight line. Let LMP be the straight line meeting the axes in the points L, M. Let OM c, and let tan OLM = m. Let x, y be the co-ordinates of any point P on the line. Draw PN parallel to the axis of y, and OQ parallel to the line LMP, as in the figure. Then But NP=NQ+QP = = ON tan NOQ + OM. NP=y, ON = x, OM= c, and tan NO Q = tan OLM = m. which is the required equation. (i) So long as we consider any particular straight line the quantities m and c remain the same, and are therefore called constants. Of these, m is the tangent of the angle between the positive direction of the axis of x and the part of the line above the axis of x, and c is the intercept on the axis of y. By giving suitable values to the constants m and c the equation y=mx+c may be made to represent any straight line whatever. For example, the straight line which cuts the axis of y at unit distance from the origin, and makes an angle of . 45° with the axis of x, has for equation y = x + 1. We see from (i) that the equation of any straight line is of the first degree. Y 16. To shew that every equation of the first degree represents a straight line. The most general form of the equation of the first degree is Ax+By+C=0........ .(i) To prove that this equation represents a straight line, it is sufficient to shew that, if any three points on the locus be joined, the area of the triangle so formed will be zero. Let (x', y'), (x", y''), and (x"", y'") be any three points on the locus, then the co-ordinates of these points will satisfy the equation (i). the area of the triangle is therefore zero [Art. 6]. The equation Ax + By + C = 0 is therefore the equa tion of a straight line. 17. The equation Ax+ By +C=0 appears to involve three constants, whereas the equation found in Art. 15 only involves two. But if the co-ordinates x, y of any point satisfy the equation Ax + By + C = 0, they will also satisfy the equation when we multiply or divide throughout by any constant. If we divide by B, we can write Ꭺ C the equation y=- X and we have only the two B C B B' which correspond to m and c in the 18. To find the equation of a straight line in terms of the intercepts which it makes on the axes. Let A, B be the points where the straight line cuts the axes, and let OA = a, and OB = b. Let the co-ordinates of any point P on the line be x, y. Y or B N Draw PN parallel to the axis of y, and join OP. This equation may be written in the form lx + my = 1, where l and m are the reciprocals of the intercepts on the axes. 19. To find the equation of a straight line in terms of the length of the perpendicular upon it from the origin and the angle which that perpendicular makes with an axis, Let OL be the perpendicular upon the straight line AB, and let OL = p, and let the angle XOL=a. Let the co-ordinates of any point P on the line be x, y. Draw PN parallel to the axis of y, NM perpendicular to OL, and PK perpendicular to NM, as in the figure. which is the required equation. 20. In Articles 15, 18 and 19 we have found, by independent methods, the equation of a straight line involving different constants. Any one form of the equation may however be deduced from any other. For example, if we know the equation in terms of the intercepts on the axes, we can find the equation in terms of p and a from the relations a cos a = =p and b sin a = which we obtain at once from the figure to Art. 19. Hence P, |