ANALYTIC GKOMETRY [Cii. XII. 



the same locus as equation (2); hence it is proved that, in 

 nvtaiiLMil.ir OOdrdinateS, </</// t-ywitinn of th<-fvrm 



Ax 9 + 2 Uxy + By* + 2Cte + 2 J FV+C:=0 



represent* a conic section whose axes are inclined at an angle 6 

 to the yiven coordinate axes, where 6 is determined by the 

 equation 



It is to be noted that the constant term C has remain* d 

 unchanged by the transformation given above. 



The next article will illustrate the application of this 

 method to numerical equations. It is to be observed that 

 this method is entirely general, and enables one to fully 

 determine the conic represented by any given numerical 

 equation of the second degree. 



NOTE. In the proof just given that every equation of the second 

 degree represents a conic section, it is assumed that the given axes are at 

 right angles. This restriction may, however, be removed; for if they are 

 not at right angles, a transformation may be'inade to rectangular axes 

 having the same origin (cf. Arts. 74, 75), and the equation \vill have its 

 form and degree left unchanged; after which the proof already given 

 applies. 



176. Illustrative examples. EXAMPLE 1. Given the equation 



-ll =0, . . . (1) 



to determine the nature and position of its locus. 



Turn the axes through an angle 0, i.e., substitute for x and y, respec- 

 tively, x 1 cos \f sin and x* sin + tf cos 0; equation (1) then becomes 



x"(-co80 + 4sin0cos0 - 



+ zy(+ 2 sin cos + 4 cos*0 - 4 sin 8 - 2 sin cos 0) 



- i/* (sin 2 + 4 sin 6 cos 6 + cos 2 6) 



- *> (4 V2 cos - 2 V2 sin 0) 



+ y'(+4\/2sin0 + 2 V2cos0)- 11 = 0. ... (2) 



