DISCRIMINATION OF THE CONIC. ,^5 



The Central Coiiie. — If ab — //- is not zero, the co-ordinates 

 of the centre of the conic are finite. Transform the equation to 

 parallel axes through the centre, and it becomes 



y = n.r- -f 2hsy + ^V" + c' = o (8) 



where 



_ _ c' =ax- + 2kxy -f ^v- + 2gx + 2fY -|- c, (9) 



X and _v being the co-ordinates of the centre. 

 The slope of the central conic is then 



dv o.r -|- /r3' _ h , li^ — ah iio 



Tzvo intersecting Straight Lines. — If c' =r o. the slope is 



double-valued, and is independent of the co-ordinates. Subject 

 to this condition, therefore, the equation becomes 



as- + 2lixy --f- by^ = 0. (11) 



and must represent two straight lines of different slopes, and 

 therefore intersecting. The condition may be written in the 

 form 



g-v + fy + c = o, (12) 



and therefore 



2fgli -f abc — af — bg- - ch-= o. (13) 



Tangent at Infinity. Hyperbola and Ellipse. — If c' does not 

 vanish, the slope is a function of the co-ordinates. The slope of 

 the tangent at infinity is, from ( 10) 



—r ' j~± j~y/lr — ah. (14) 



dx -V —^00 b h ^ ^ ^' 



]f /;'- > ab, these values are real, and the conic is a hyperbola; 

 if h'- < ab, the}- are imaginary, and the conic is an ellipse. 



Axes of a Central Conic. — The method of (i) is again 



a|)plied to determine the axes of a central conic, .9' = o. \\^ith 

 the same centre describe a circle — 



C = x^^y^ = r\ (15) 



The slope of the conic is 



dy ax -\- hy 



dx^ hx -{- by 



The slope of the circle is 



dy _ X 



dx„ ~~ y. 

 The locus 



d\ dv 



(16) 



(17) 



(18) 



c/.v, dx^ 



or, 



hy- — hx- = xy(b — a) (19) 



according to ( 1 1 ) represents two intersecting straight lines pass- 

 ing through the origin, and from (18) they must be lines passing 



