Plane and solid analytic geometry; an elementary textbook . P1KP2 we haveLP KP Fig. 27. PXL PXK But LP = y- yxPXL = x — xvKP2 = y2- yvP1K=x2 -xv 34 Ch. IV. §23] THE STRAIGHT LINE 35 Substituting these values, we have y -V\ = Vi-y\t 1-5-1 This is then the algebraic relation between the coordi-nates x and y of any point on the line and the constantsxv yv x2, and y2, and is therefore the equation of theline. It is called the two-point form of the equation ofthe straight line. Let the student showthat this equation cannotbe satisfied by the coor-dinates of any point noton the line. The student
Plane and solid analytic geometry; an elementary textbook . P1KP2 we haveLP KP Fig. 27. PXL PXK But LP = y- yxPXL = x — xvKP2 = y2- yvP1K=x2 -xv 34 Ch. IV. §23] THE STRAIGHT LINE 35 Substituting these values, we have y -V\ = Vi-y\t 1-5-1 This is then the algebraic relation between the coordi-nates x and y of any point on the line and the constantsxv yv x2, and y2, and is therefore the equation of theline. It is called the two-point form of the equation ofthe straight line. Let the student showthat this equation cannotbe satisfied by the coor-dinates of any point noton the line. The student shouldhere, and in all the fol-lowing demonstrations,assure himself that theproof is perfectly general. Place the lines and points indifferent positions, being careful to give the same letterto corresponding points, and the demonstrations ought tohold, letter for letter. For example, try Fig. 28 with theabove demonstration, being careful to note that PXL = Mx0 + OM = OM - 0MVLP = LM + MP =MP - ML,P1K=3I10-{- 0M2 = 0M2 - 0MV. KP, = KM, + M9PK M2P2 - M2K. Equation [5] may be written in the determinate formx% y 1 = o. 3G ANALYTIC GEOMETRY [Ch. IV, §§ 24, 25 24. Line determined by its intercepts. — If the twogiven points should be, in particular, the points where theline cuts the axes, or if, in other words, the intercepts aand b are given, the equation can be found easily by sub-stituting («, 0) for (xv y^) and (0, 5) for (#2, y^) inequation [5]. It becomes y _ Q = b - 0x — a 0 — a or reducing, —+ ? = 1. [6] This is called the intercept form of the equation of thestraight line. Let the student derive equation [6] geometrically with-out using equation [5]. 25. Oblique coordinates.—In obtaining these equationsof the straight line we have made no use of the fact thatthe axes are perpendicular. The 011I3- idea used was thesimilarity of triangles, which will be true in oblique aswell as rectangular coordinates. The results will holdtherefore for both systems of Cartesi
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