H II I UK L" . XQUAT10X i ' 



oonoeraed chiefly with algebraic curve* of the first and 



37. Construction of loci. Discussion of equations. The 

 process of const m< -ung & locus ly plotting separate point*, 

 tii'-n oo ' them by a smooth curve, is only ap- 



<, .in. I IN 1.-M- ami tedious. It may of teii be short- 

 ened by a consideration of the p H of the given 

 equa . h as syiuni- limiting values of the vari- 

 ables fur \\lu.-li lx>th are real, etc. Such considerations will 

 i .sh<>\\ tin- general form and limitation*, of tin- mrve ; 

 ami, t.iUn together, they ooostiteto eHMnnfMi :'" ' ; " 



The points where a locus crosses the coordinate axes are 

 almost always useful ; in <lru \\rng the curvr, they are given 

 l.\ thru <listances from the origin along the respective axes. 

 These distances are called the intercepts of the curve. 



The following examples may serve to illustrate these 

 concept i 



(1) Ditcuuion oftki equation 8x - 2y + 12 = [see (.1) Art S3]. 



Interoepta : if r - 0. then jf = 0; hence the y-intercept is 6 

 (aee Fig. 16) ; if y = O t then x = - 4 ; hence the '-intercept is 4. 



The equation may be written : z = ) y - 4, which shows that as y 

 iaoreasM continuously from to co , x increases continuously f r. .in - 4 

 to oo ; therefore the locus passes from the point P t through the point /',. 

 and then recedes indefinitely from both axes in the first quadrat 

 ten as aboTe, the equation also shows that as y decreases from to - * . 

 * also decreases from - 4 to - co ; therefore the locus passes from /'. 

 the third quadrant, receding again indefinitely from both axes. Sines 

 for erery value of y, z takes but one ralue (i>.. each value ot y corre- 

 sponds to but one point on the curve), therefore the locus consist* of a 

 le branch. The prnnf that the locus of any fimt-degree equation, in 

 two variables, is a straight line is given in Chap. V. 



/>iViuiV>n oftHr ^uation y* = 4 x. [See (4) Art. 33.] 



Fig. 17): if x = 0, then f = 0,and if y = 0, then * =C; 



AH. OBOM. 4 



