. Plane and solid analytic geometry; an elementary textbook. Draw MN through P1 parallel to AB and continue the perpendicular Off to meet it at K. The equation of MN will be x cos u + y sin a — px = 0, where px may be either positive or negative. For, as thevalue of a is fixed and as MN can be any line parallel toAB, it may be on the opposite side of the origin from AB,and in this case px will be negative. (See Art. 34.) Since Px lies on MN, its coordinates (xv y-^) must satisfythe equation of MN. 54 ANALYTIC GEOMETRY [Ch. IV. § 36 Hence xx cos a + yx sin « = pv Now wherever P1 may lie, RPX =
. Plane and solid analytic geometry; an elementary textbook. Draw MN through P1 parallel to AB and continue the perpendicular Off to meet it at K. The equation of MN will be x cos u + y sin a — px = 0, where px may be either positive or negative. For, as thevalue of a is fixed and as MN can be any line parallel toAB, it may be on the opposite side of the origin from AB,and in this case px will be negative. (See Art. 34.) Since Px lies on MN, its coordinates (xv y-^) must satisfythe equation of MN. 54 ANALYTIC GEOMETRY [Ch. IV. § 36 Hence xx cos a + yx sin « = pv Now wherever P1 may lie, RPX = HK= OK— OIT—p1—p. Hence BPi = xx cos a + yx sin a -p. [IT, a] If the equation is given in the form Ax + By +(7=0,it is necessary first to reduce it to the normal form andthen substitute x1 for x and y1 for y. Hence BPL = ^ ? [17,5 ] 7%e radical must be given the sign opposite to that of C. It appears from the way BPX has been chosen that theresult will be positive when the point and the origin are onopposite sides of the line ; negative, when t
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