304 AffAi. Y'l'ir <-/-.o .\ii:n;y xn. 



The coordinates of the center are a= - 1, 0=3. 

 Therefore, C=C7a+F+C= -4; /i' + tf = 6, -4'ff=-8; 

 and, since A' is larger than /?, // being positive (Art. 130), 

 hence A' = 4, 7^ = 2; 



whilo tan 2 = oo , and therefore 0=45. The transformed equation is 

 therefore 



when referred to the axes IPX", &Y"; and the locus is approximately as 

 given in Fu. r."J. 



b. Non-central conic. If // 2 AB = 0, the relations of equations (<') 

 and (11), Art. 180, may still be used to simplify the reduction of equa- 

 tion (1) to the standard form for the equation of a parabola, if, as in 

 Art. 170, the xy-term be removed first. In this case, however, a better 

 method of reduction is as follows : 



Since the first three terms of equation (1) form a perfect square, that 

 equation may be written 



therein the sign of the VB is the same as that of //. 



Equation (2) may now be transformed to new axes OX' and 0V, 

 which are so chosen that the equation of OX' referred to the given axes 

 shall be 



VAx+ 



hence, if 6 be the angle between OX and OX', then 



whence sin = -^L and cos = -__ . (7) 

 ~ 



Equation (7) shows that is negative (if the positive value of VA + B 

 be used), and acute or obtuse according as VB is positive or negative. 

 The formulas for transforming to the new axes are (cf. Art. 7J) 



Substituting these values for z and y in equation (6), it becomes 



* V3 .'.|.a ^ + fVB > ' + C = 0. . (9] 



VA +B -SA + B 



