. Algebraic geometry; a new treatise on analytical conic sections . FlQ. SI. PN - QN changes its sign, which proves the proposition. 36 THE STRAIGHT LINE. [chap. II. When P is at the origin, Kj = y^ = 0, and then Aajj + B^i + C = C. .. if C and the expression ASj + By, + C have the same sign,the point (xj, y-^ and the origin are on the same side of the line. If C and Aajj + Byj + C have opposite signs, the point {x-^, y^)and the origin are on opposite sides of the line. Consider the straight line 3x + iy-l2 = 0. Here C = - 12. Nowwhen a! = 3 and y = 4, 3a; + 4y-12 = 9 +16-12 = 13,which is posi
. Algebraic geometry; a new treatise on analytical conic sections . FlQ. SI. PN - QN changes its sign, which proves the proposition. 36 THE STRAIGHT LINE. [chap. II. When P is at the origin, Kj = y^ = 0, and then Aajj + B^i + C = C. .. if C and the expression ASj + By, + C have the same sign,the point (xj, y-^ and the origin are on the same side of the line. If C and Aajj + Byj + C have opposite signs, the point {x-^, y^)and the origin are on opposite sides of the line. Consider the straight line 3x + iy-l2 = 0. Here C = - 12. Nowwhen a! = 3 and y = 4, 3a; + 4y-12 = 9 +16-12 = 13,which is positive, while C( - 12) is negative. .. the point (3, 4) and the origin lie on opposite sides of theline 3a; + 4y-12 = 0. When a;=2 and j^=-1, 3a; + 42/-12 = 6-4-12= - 10. .. the point (2, - 1) and the orjgin lie on the same side of theline. 38. The length of the perpendicular from the point (iSj, y-^) upon thefiiX+By + C = 0 is equal to AXi + Byi + C y. Let EF be the straight line, P the point (aij, y.^.Draw PK perpendicular to EF. Also draw PN perpendicular to Qx, meeting EF at u OEF = a. At E, ^ = 0; .. Aa; + C = 0 and x= --, 0E= --. AM. 39.] THE STRAIGHT LINE. 37 In the same way at F, a; = 0 and 0F= - - ; B .. tan a = —- = - and sin a = Also L KPQ— 90° - L KaP = 90° -L NQE = a ; .■. PK=PQcosa = (PN-QN)cOSa = ^^^L==SSl£. (1) But Q is on the line Aa; + Bj/ + C t= 0, and (Kj, QN) are itsco-ordinates; /. Aa;i + + C = 0, whence QN=-^^^. B / , A!j;,+C\^ from (1) PK = iLl_=Lj! = ^^l±ll±^. Corollary. When P is at the origin, the expression for theperpendicular becomes C x/A2+B2 AiU 4 Bl/ 4- CBut when P crosses the line, the expression —^, ^ ^ ^—changes its sign (Art. 37). VA + B .. if P and the origin are on the same side of the line, the C , Aaii + Byi + C , ,, . , expressions , ^ ^ and — , - have the same sign; and when P and the origin are on opposite sides of the line, theseexpressions have opposite si
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